Patients must have achieved a radiologic complete or partial remission as defined by the Lugano criteria Prior or concurrent malignant disease unless in complete remission for more than five years Known history of another primary malignancy that has not been in remission for ? year Biopsy-proven intermediate or high-grade non-Hodgkins lymphoma, meeting one of the following criteria (timeline months prior to enrollment):\r\n* In partial remission\r\n* Relapsed after initial complete remission\r\n* Failed induction therapy, but responds to salvage therapy (i.e., chemosensitive disease)\r\n* In complete remission with high-risk features as specified by the International Prognostic Index Biopsy-proven advanced stage mantle cell lymphoma with proliferation-related Ki- antigen (Ki-) > % in first complete remission (timeline months prior to enrollment) Biopsy-proven Hodgkins lymphoma, meeting one of the following criteria (timeline months prior to enrollment)\r\n* In first, or greater relapse after initial complete remission\r\n* In partial remission\r\n* Failed induction therapy, but responds to salvage therapy (i.e., chemosensitive disease) Biopsy-proven Burkitts lymphoma, meeting one of the following criteria (timeline months prior to enrollment):\r\n* In second complete remission after relapse following initial complete remission\r\n* Failed induction therapy, but responds (very good partial remission, complete remission, or near complete remission) to salvage therapy (i.e., chemosensitive disease) NOTE: Patients meeting the following criteria are exempt from the month timeline and do not require additional biopsy:\r\n* Patients who have never achieved a complete remission on the last biopsy-proven site of disease and went on to the next therapy then achieved a complete remission\r\n* Patients who relapsed quickly (within months of their last chemotherapy) and now have achieved a complete remission with salvage therapy Complete second remission (CR). Complete second remission (CR). Any remission after chimeric antigen receptor (CAR)-T cell therapy AML that has failed to achieve complete remission or morphologic complete remission or For AML: patients must belong to one of the following high risk categories: \r\n* Primary induction failure (PIF) as defined by failure to achieve at least a % reduction in bone marrow blasts after one cycle of high intensity, anthracycline containing induction regimen or failure to achieve complete response (CR)/complete remission with incomplete blood count recovery (CRi) after two cycles of high intensity chemotherapy\r\n* First early relapse as defined by an initial remission duration of fewer than months\r\n* Second or subsequent relapse regardless of remission duration, or\r\n* Relapse after allogeneic or autologous stem cell transplantation (first relapse after stem cell transplant would be eligible, regardless of prior duration of remission) First relapse if first remission ? months Relapsed or refractory (resistant) disease, as defined by standard criteria\r\n* Relapsed: Bone marrow blasts >= %, reappearance of blasts in the blood, or development of extramedullary disease following achievement of complete remission (CR)/complete remission with incomplete recovery of blood counts (CRi)/complete remission with incomplete recovery of platelets (CRp)/morphologic leukemia free state (MLFS)\r\n* Refractory (resistant): Failure to achieve CR/CRi/MLFS in subjects who survive >= days following completion of initial treatment, with evidence of persistent leukemia by blood and/or bone marrow examination Patients without evidence of documented disease progression clinically or radiographically after ASCT (stable disease [SD], partial remission [PR] or complete remission [CR]) who have had count recovery (absolute neutrophil count [ANC] > cells/mm^, non-transfused platelet count > , K/mm^) and are at least days post ASCT but no more than days post ASCT Patients in partial or complete remission following cell therapy will also be eligible NHL patients with resistant or refractory lymphoma (no partial remission [PR] following up to three cycles of combination chemotherapy) will not be eligible for transplant in this trial Lymphoblastic lymphoma\r\n* All patients will be eligible in second or greater complete remission (CR) or first or subsequent partial remission (PR)\r\n* Patients with any high-risk features will be eligible in first complete remission\r\n* High risk features include:\r\n** Stage IV\r\n** Lactate dehydrogenase (LDH) > x upper limit of normal\r\n** >= extranodal sites PHASE II: Patients who have/are either:\r\n* Transplanted in hematologic first complete remission with evidence of minimal residual disease within days of allogeneic transplantation\r\n** Post-Transplant Minimal Residual Disease defined by:\r\n*** Any detectable ALL (by flow cytometry, cytogenetics, or PCR techniques) as per clinical indication\r\n* In second or third complete remission at the time of allogeneic transplantation\r\n* Treated with reduced intensity regimens\r\n* Lymphoid blast crisis of CML\r\n* Are relapsed or refractory to at least line of chemotherapy In a morphologic complete remission with incomplete blood count recovery, or morphologic complete remission post-induction after no more than induction cycles Relapsed T-cell lymphoma that is chemotherapy sensitive in CR/partial remission (PR) that has failed or ineligible for an autologous transplant Plasma Cell Leukemia after initial therapy if achieved at least in partial remission; or relapsed and achieved subsequent remission (CR/PR) Have a newly diagnosed AML, based on World Health Organization criteria, currently in first (st) complete remission (CR)/complete remission with incomplete count recovery (CRi) on a bone marrow biopsy performed within weeks of study enrollment Patient relapsing more than years after initial remission History of malignant tumors other than KS or KSHV-MCD, unless:\r\n* In complete remission for >= year for the time complete remission was first documented\r\n* Resected basal cell or squamous cell carcinoma of the skin\r\n* In situ cervical or anal dysplasia Patients must be in a >= nd complete remission as indicated by appropriate radiologic evaluations at the time of study entry Patients with OS in first complete remission Acute lymphocytic leukemia\r\n* In complete first remission (CR) in addition to one of the criteria outlined\r\n* Second or greater complete remission Disease status must be complete remission by standard criteria for lymphoma and acute leukemia patients Patients who have undergone an autologous HCT for the treatment of DLBCL and are in a complete remission (CR), partial remission (PR) or have stable disease (SD) at the day post-transplant reassessment Patients aged to years with high risk AML who have achieved their FIRST complete remission (CR) or complete remission with incomplete recovery (CRi) within months of enrollment and are not immediately candidates for allogeneic stem cell transplant; patients above age who are not eligible for other protocols may be considered for enrollment on a case by case basis after discussion with the principal investigator (PI) Acute lymphocytic leukemia or acute myelogenous leukemia who are not in first remission or second remission i.e. after failing induction therapy, or in relapse or beyond second remission; (prior therapy with VP- and Cytoxan is allowed) Patients in complete remission with no assessable disease Patients with hematologic malignancies for whom autologous stem cell transplantation is deemed clinically appropriate:\r\n* Non-Hodgkins lymphoma, or Hodgkins lymphoma: either in a first complete remission (CR) or refractory/relapsed with chemosensitive disease in a complete remission (CR) or partial remission (PR)\r\n* Multiple myeloma in first or second remission; patients with CR or PR will be eligible for this protocol Note: Patients who have received transplant during st remission are excluded since this would be considered a nd treatment PHASE I: Diagnosis of a hematological malignancy listed below (excluding myelofibrosis) in remission or with stable minimal residual disease\r\n* Acute myelogenous leukemia (AML) in st or subsequent remission or in relapse after any remission\r\n* Acute lymphoblastic leukemia (ALL) in st or subsequent remission or in relapse after any remission\r\n* Myelodysplastic syndrome either intermediate or , or high risk by the International Prognostic Scoring System\r\n* Chronic myelogenous leukemia (CML) in accelerated or second chronic phase\r\n* Non-Hodgkins lymphoma (NHL) or Hodgkins disease (HD) in nd or greater complete remission, partial remission, or refractory relapse\r\n* Chronic lymphocytic leukemia (CLL), Rai stage -, failing at least prior regimens\r\n* Multiple myeloma (MM), stage -\r\n* Myeloproliferative disorder or neoplasm Any patient, regardless of age or sex, with EBV-positive Hodgkins or non-Hodgkins lymphoma, (regardless of the histological subtype) or EBV (associated)-T/NK-lymphoproliferative disease or severe chronic active EBV (CAEBV); in second or subsequent relapse (or first relapse or with active disease if immunosuppressive chemotherapy contraindicated or multiply relapsed patients in remission who have a high risk of relapse) OR any patient with primary disease or in first remission if immunosuppressive chemotherapy is contraindicated, e.g. patients who develop Hodgkin disease after solid organ transplantation or if the lymphoma is a second malignancy e.g. a Richters transformation of chronic lymphocytic leukemia (CLL) (Group A); OR in remission or with minimal residual disease status after autologous or syngeneic stem cell transplantation (SCT) (Group B) Patients in first remission are eligible Standard risk patients will include eligible patients, as defined above, who are receiving transplants as treatment for MDS in RA/RCMD, RARS, AML in st or nd remission, ALL in st CR, NH in st remission, MM in st remission, very good partial response, or st partial response on the CML in the first chronic phase or st remission Acute leukemias in second (nd) or subsequent remission High-risk NB (as defined above) and in ) first CR/VGPR at >= months from initiation of immunotherapy using anti-GD antibody, or ) second or subsequent remission; remission is defined as complete (CR) or very good partial (VGPR) remission, according to the International Neuroblastoma Response Criteria; urine catecholamine levels are no longer taken into consideration when staging; patients can be considered as in VGPR with or MIBG (+) sites that were previously-irradiated All children and adults with AML who have achieved a first or second bone marrow remission are eligible for this protocol; patients must undergo peripheral blood stem cell collection or marrow harvest while in remission and must not be expected to have better outcomes with allogeneic transplantation Chronic lymphocytic leukemia (must have all three): Rai Stage III/IV; progression after previous complete remission (CR) or partial remission (PR) including purine antagonist (i.e. fludarabine); recent chemotherapy responsiveness Non-Hodgkin's lymphoma (NHL) in second or third complete remission, or relapse (including relapse post autologous hematopoietic stem cell transplant); double hit lymphomas in first remission or more advanced disease History of lymphoma (Richters syndrome) unless in complete remission > years without\r\nrelapse Patients with current active malignancies or any remission for < months, except patients with carcinoma in situ or with non-melanoma skin cancer who may have active disease or be in remission for less than months Subjects must be in primary remission NOTE: All patients achieving CR or complete remission with incomplete blood count recovery (CRi) will receive consolidation when fit Relapsed after achieving remission with a prior therapy Non-Hodgkin or Hodgkin lymphoma without active central nervous system (CNS) involvement associated with poor prognosis with medical therapy alone or for which autologous peripheral blood stem cell (PBSC) transplant is indicated:\r\n* Hodgkins lymphoma beyond first remission or first partial remission or induction failure with subsequent response to salvage therapy\r\n* Non-Hodgkins lymphoma beyond first remission or first partial remission or induction failure with subsequent response to salvage therapy\r\n* Chemotherapy responsive disease Diagnoses to be included:\r\n* Acute Myelogenous Leukemia at the following stages:\r\n** First remission (cytogenetic intermediate or high risk)\r\n** Second or subsequent remission\r\n*** Complete remission is defined as the absence of blasts in the peripheral circulation at the time of enrollment and < % blasts in the bone marrow\r\n* Chronic Myelogenous Leukemia at the following stages:\r\n** First or subsequent chronic phase:\r\n*** Patient refused tyrosine kinase therapy or is otherwise not suited for it\r\n*** Patient who has failed two lines of tyrosine kinase therapy (e.g., patient has not had a complete hematologic response and/or minor cytogenetic response by months of second line therapy, major cytogenetic response by months of second line treatment, or complete cytogenetic remission (CCyR) by months of second line treatment)\r\n*** Patient who has lost complete hematologic response or major/complete cytogenetic response while on second line of therapy\r\n** Accelerated Phase any one of the following symptoms:\r\n*** White blood cell (WBC) difficult to control (> x ^/L despite therapy)\r\n*** Rapid doubling of WBC (< days)\r\n*** % blasts in blood or marrow\r\n*** % blasts and/or promyelocytes in blood or marrow\r\n*** % basophils and/or eosinophils in blood\r\n*** Anemia or thrombocytopenia unresponsive to standard treatment\r\n*** Persistent thrombocytosis (> x^/L)\r\n*** Cytogenetic abnormalities in addition to Philadelphia positive (Ph+)\r\n*** Increasing splenomegaly\r\n*** Marrow fibrosis\r\n* Myelodysplastic syndromes at any of the following stages:\r\n** Refractory anemia\r\n** Refractory anemia with ringed sideroblasts\r\n** Refractory cytopenia with multilineage dysplasia\r\n** Refractory cytopenia with multilineage dysplasia and ringed sideroblasts\r\n** Refractory anemia with excess blasts- (-% blasts)\r\n** Refractory anemia with excess blasts- (-% blasts)\r\n** Myelodysplastic syndrome, unclassified\r\n** MDS associated with isolated del (q) (only after failing lenalidomide)\r\n** Low risk MDS patients would be eligible only if transfusion-dependent and failing standard therapy (i.e. hypomethylating agents)\r\n** Chronic Myelomonocytic Leukemia\r\n* Primary Myelofibrosis\r\n** Intermediate- risk or high risk disease\r\n** Patients should have extinguished standard of care options prior to being considered for this trial\r\n* Chronic Lymphocytic Leukemia\r\n** Complete remission: the disease is completely absent and no relapse occurred prior to the preparative regimen; requires all of the following: \r\n** Nodular partial remission: complete response with persistent lymphoid nodules in bone marrow\r\n** Partial remission: reduction of more than % in the disease burden regardless of the number of lines of therapy received\r\n** Eligibility will be limited to those who have at least failed a fludarabine-based regimen\r\n* Mature B cell malignancies (including mantle cell lymphoma, follicular lymphoma, diffuse large B cell lymphoma, non-Hodgkin lymphoma not otherwise specified)\r\n** Patients should have extinguished standard of care options prior to being considered eligible for this trial\r\n** First complete remission (CR) confirmed: complete disappearance of all known disease; the term confirmed is defined as a laboratory and/or pathological or radiographic determination\r\n** CR unconfirmed (CRU): complete disappearance of all known disease with the exception of persistent scan abnormalities of unknown significance; the term unconfirmed is defined as scan abnormalities of unknown significance that are not biopsied or otherwise evaluated\r\n** Second complete remission positive (CR+) confirmed: the recipient relapsed, then achieved complete absence of disease without radiographic evidence of disease\r\n** CR+ unconfirmed: the recipient has achieved a second or subsequent complete response but has persistent radiographic abnormalities of unknown significance\r\n** Partial remission: reductions of >= % in greatest diameter of all sites of known disease and no new sites Subjects who have an uncontrolled systemic malignancy that is not in remission Mantle cell lymphoma (MCL), T-NHL, or other high-risk malignancies may be enrolled/transplanted in complete remission (CR)/first partial remission (PR) NHL patients with resistant or refractory lymphoma (no partial remission [PR] following up to three cycles of combination chemotherapy) will be ineligible for transplant in this trial Lymphoblastic lymphoma \r\n* All patients will be eligible in second or greater complete remission (CR) or first or subsequent partial remission (PR)\r\n* Patients with any high-risk features will be eligible in first complete remission \r\n* High risk features include: \r\n** Stage IV\r\n** Lactate dehydrogenase (LDH) > x normal\r\n** >= extranodal sites Burkitts/Burkitts like\r\n* All patients except localized lymphoma will be eligible anytime after initial therapy (after achievement of first complete remission), or in partial remission if they fail to achieve CR\r\n* Patients with localized (stage I or Ziegler stage A) will be eligible only if they fail to achieve CR or after relapse Chronic lymphocytic leukemia\r\n* Relapse post-fludarabine\r\n* Non-complete remission (CR) after salvage regimen Subject did not achieve complete remission/complete remission with incomplete hematologic recovery/complete remission with incomplete platelet recovery (CR/CRi/CRp) under initial therapy. A subject eligible for standard therapy must receive at least one cycle of an anthracycline containing induction block in standard dose for the selected induction regimen. A subject not eligible for standard therapy must have received at least one complete block of induction therapy seen as the optimum choice of therapy to induce remission for this subject. A primary malignancy which has been completely resected and in complete remission for ? years First Complete remission (CR)/ Complete remission with incomplete blood count recovery (CRi) with induction therapy + consolidation therapy within months (+/- days of achieving CR or CRi) Participants with hematologic malignancies or hematologic disorders for whom allogeneic stem cell transplantation is deemed clinically appropriate; eligible diseases and stages include:\r\n* Non-Hodgkin's lymphoma, or Hodgkin's lymphoma in second (nd) or subsequent complete remission or in partial remission with documented chemosensitivity to the most recent chemotherapy regimen; prior autologous transplantation is required, unless deemed medically inappropriate by the treating physician\r\n* Multiple myeloma: relapsed but with chemosensitive disease; bone marrow plasma cells may not exceed % of the total cellularity\r\n* Chronic lymphocytic leukemia: any Rai stage III or IV, lymphocyte doubling time of months, or stage I-II with progression after >= chemotherapy regimens, in partial remission with documented chemosensitivity to the most recent chemotherapy regimen\r\n* Acute myelogenous or acute lymphoblastic leukemia in second or subsequent complete remission or in first remission with adverse cytogenetic/molecular features or a documented antecedent hematologic disorder\r\n* Myelodysplastic disorder\r\n* Myeloproliferative disorder including myelofibrosis, chronic myelogenous leukemia resistant to tyrosine kinase inhibitors\r\n* Aplastic anemia with no response to immunosuppressive therapy Patients with a history of non-hematologic malignancies (except non-melanoma skin cancers) currently in a complete remission, who are less than years from the time of complete remission, and have a > % risk of disease recurrence Patients in complete remission with no evidence of evaluable disease by radiologic imaging Patients with multiple myeloma in complete remission (CR), partial remission (PR), or very good partial remission (VGPR), or symptomatic stable disease (no evidence of progression) including patients with light chain MM detected in the serum by free light chain assay Participants must have chemosensitive disease prior to ASCT, defined as achieving at least a partial remission (as determined with PET imaging) to salvage treatment using International Harmonization Project (IHP) criteria; participants with cHL or DLBCL (arms A and B) transplanted in first (st) remission after only one line of treatment are not eligible; participants with PTCL (arm C) transplanted beyond st remission are also not eligible Patient is currently in complete cytogenetic remission (CCyR) Patients will have relapsed at least once and returned to complete clinical remission after additional chemotherapy; interval surgery is permitted Patients with non-T-cell-based lymphoma of any type or hairy cell leukemia are eligible on the condition that they do not receive active systemic treatment for their hematologic disease and are in complete remission as evidenced by PET/CT scans and bone marrow biopsies for at least months. Acute myelogenous leukemia (AML) in complete morphological remission at study screening (Complete Remission with Incomplete Platelet Recovery (CRp) acceptable). Acute lymphoblastic leukemia (ALL) in complete morphological remission at study screening (Complete Remission with Incomplete Platelet Recovery (CRp) acceptable). Patients who have had a previous malignancy that is not in remission Patients must have had a previous auto-SCT performed as part of a consolidation of an initial remission and had a remission, defined as a partial response or greater that lasted at least months either on or off maintenance therapy without evidence of progression as defined by IMWG criteria Patients must have partial remission (PR) to salvage chemotherapy Patients not in remission must have CD-expressing leukemic blasts; patients in remission do not require phenotyping and may have leukemia previously documented to be CD negative (because in remission patients, virtually all antibody binding is to non-malignant cells which make up >= % of nucleated cells in the marrow) Underlying cancer in remission Patients who have had a previous malignancy that is not in remission Second primary malignancy that has not been in remission for greater than years. Exceptions that do not require a year remission include: related non-melanoma skin cancer or resected melanoma in situ. Prior malignancy (excluding nonmelanoma carcinomas of the skin) unless in remission for years. Patients must be in complete remission post transplant Acute lymphocytic leukemia with induction failure, first complete remission with high risk cytogenetics (e.g. Philadelphia positive chromosome, t[:]; remission requiring more than chemotherapy to achieve remission, or any stage beyond CR Patients not in remission must have cluster of differentiation (CD)-expressing leukemic blasts; patients in remission do not require phenotyping and may have leukemia previously documented to be CD negative (because in remission patients, virtually all antibody binding is to non-malignant cells which make up >= % of nucleated cells in the marrow) Acute myelogenous leukemia\r\n* In first complete remission with high-risk cytogenetics\r\n* Primary induction failure\r\n* In second or greater complete remission\r\n* Secondary AML\r\n* In first complete remission with hyperleukocytosis at diagnosis Acute lymphocytic leukemia\r\n* First complete remission, with high-risk cytogenetics\r\n* Primary induction failure\r\n* Second or greater complete remission Non-Hodgkin's lymphoma (NHL) in second or third complete remission, or relapse (including relapse post autologous hematopoietic stem cell transplant); double hit lymphomas in first remission or more advanced disease; or Relapse after achieving initial remission or failure to achieve initial remission Adequately treated Stage I cancer from which the subject is currently in remission and has been in remission for ? years Patient should have a responsive skin disease including complete remission (CR) and partial remission (PR) (close to CR; %-% clearance of skin disease from baseline without new tumors (T) in patients with T, T or T only skin disease) and should not have visceral organ or lymph node involvement prior to transplantation Bone marrow in morphologic remission (any remission number) defined as < % blasts (M classification) performed in local institution lab History of a concurrent or second malignancy, except for specified exceptions in the protocol or any other cancer that has been in complete remission for ? years Acute lymphoblastic leukemia \r\n* >= second complete remission (CR) (adults >= years and =< years) \r\n* CR in pediatrics (defined as < years) and < months duration of first remission\r\n* >= third complete remission (CR) or not in remission (pediatric patients < years)\r\n* T cell leukemia >= CR\r\n* Evidence of pre-transplant minimal residual disease (MRD) by flow cytometry, fluorescence in situ hybridization (FISH) or cytogenetics Individuals who have undergone transplant for hematologic malignancy are required to be in complete remission. Individuals with PMF, post-PV MF, or post-ET MF who are receiving ruxolitinib and meet Revised International Working Group for Myelofibrosis Research and Treatment (IWG-MRT) and European Leukemia Net (ELN) response criteria with progressive and relapsed disease, with modifications for progressive disease complete remission (CR), partial remission (PR), or clinical improvement (CI) Relapsed after achieving remission with a prior therapy Prior or concurrent invasive malignant disease, unless in complete remission for more than three years. Patients will be considered eligible for the study if after transplant they achieved hematologic (< % blasts) and cytogenetic remission Underlying primary disease, for which the subject underwent transplant, is in morphologic remission No plans for additional post-remission chemotherapy. Relapsed, refractory, or progressive acute leukemia in second or subsequent remission, with remission defined as < % bone marrow blasts morphologically Partial Remission (PR) Morphologic complete remission with incomplete blood count recovery (CRi) AML in second or greater remission, primary induction failure and patients with relapsed disease, peripheral blood blasts < /microliter; patients > years and =< years need to be in morphologic remission at transplant (< % blasts) The patient is in morphologic leukemia-free state or in morphologic complete remission with incomplete blood count recovery (CRi). Subject has other prior malignancy that is not in complete remission. Patients who are in complete remission or have a stable disease Are in breast cancer remission with no detectable disease present Be in a complete remission. Undergoing active treatment or in remission and undergoing active surveillance All patients must be in complete remission (CR): Diagnosis of acute leukemia (AML/ALL) or advanced MDS (intermediate- [INT-] or high risk) in complete remission (complete remission [CR]/composite complete remission [CRc]/complete remission with incomplete hematologic recovery [CRi]) documented by bone marrow biopsy done within days prior to the initiation of conditioning regimen Failure to achieve any objective response (CR - complete remission, PR - partial remissino, mCR - marrow complete remission, or HI - hematologic improvement), but no evidence of disease progression within the weeks leading to the subject's first dose of IP (INVESTIGATIONAL PRODUCT) in this study (Cycle , Day ) Any patient, regardless of age or sex, with EBV-positive Hodgkins or non-Hodgkins Lymphoma, or lymphoepithelioma/leiomyosarcoma regardless of the histological subtype or EBV (associated)-T/NK-lymphoproliferative disease or severe chronic EBV \r\n* In second or subsequent relapse (or first relapse or with active disease if immunosuppressive chemotherapy contraindicated or multiply relapsed patients in remission who have a high risk of relapse) OR any patient with primary disease or in first remission if immunosuppressive chemotherapy is contraindicated, e.g. patients who develop Hodgkin disease after solid organ transplantation or if the Lymphoma is a second malignancy e.g. a Richters transformation of chronic lymphocytic leukemia (CLL) (Group A) OR\r\n* In remission or with minimal residual disease status after autologous or syngeneic stem cell transplantation (SCT) (Group B) OR\r\n* In remission or with detectable disease after allogeneic SCT (Group C) Participants must have pathologically confirmed acute myeloid leukemia (AML) in first complete remission (CR) as defined by:\r\n* Bone marrow biopsy with < % blasts\r\n* No clusters or collections of blast cells\r\n* No extramedullary leukemia\r\n* Absolute neutrophil count >= /uL (achieved post-induction at some point)\r\n* Please note that full platelet recovery is not necessary, and thus, patients achieving pathological complete remission (CRp) are eligible Diagnosis of pre-B cell or T cell ALL and in continuous first remission; at least months into maintenance up until the first year off therapy Previously diagnosed of ALL and currently in remission Patients who have achieved complete remission of intestinal metaplasia (CR-IM) In remission (complete remission [CR], partial remission [PR], or stable disease based on clinical, not necessarily radiologic, assessment) and currently being observed and with no current cytotoxic chemotherapy planned; patients may be on rituximab maintenance