REGISTRATION STEP 2-RANDOMIZATION: Patients must have morphologically confirmed, previously untreated acute myeloid leukemia (AML) or MDS with excess blasts-2 (MDS-EB-2)\r\n* Patients with acute promyelocytic leukemia (APL), biphenotypic leukemia, blastic transformation of chronic myelogenous leukemia (CML or BCR/ABL), are not eligible\r\n* Patients must have disease present in the blood or bone marrow; patients with only extramedullary disease in the absence of bone marrow or blood involvement are not eligible\r\n* All tests for establishing baseline disease status eligibility must be based on blood and/or bone marrow examination performed within 42 days prior to randomization (registration Step 2) Patients with acute bilineal/biphenotypic leukemia leukemia in 2nd or higher relapse, Patients must have a diagnosis of relapsed or refractory CD22-positive acute leukemia including B-ALL, mixed phenotype leukemia (biphenotypic), or Burkitt’s leukemia based on World Health Organization (WHO) classification; patients with bilineal leukemia are excluded Acute lymphocytic leukemia (ALL): must have < 5% marrow blasts at the time of transplant Diagnosis of acute promyelocytc leukemia Have current acute leukemia. Diagnosis of acute leukemia or any patient that has been treated with fludarabine. T-cell prolymphocytic leukemia Prior history of acute leukemia or AML NK cell leukemia. Any acute leukemia with marrow aplasia or without adequate count recovery. Mixed phenotype acute leukemia MRD > 1% after consolidation Participants must have a diagnosis of AML or acute leukemia of ambiguous lineage (acute undifferentiated leukemia or mixed phenotype acute leukemia) and meet the criteria below:\r\n* Refractory leukemia, defined as persistent leukemia after at least two courses of induction chemotherapy, OR\r\n* Early relapsed leukemia, defined as the re-appearance of leukemia after the achievement of remission and within one year of diagnosis, OR\r\n* Relapsed leukemia that is refractory to at least one course of salvage therapy (i.e., therapy given after the relapse has occurred), OR\r\n* Second or greater relapse\r\nPatients in all categories above must have >= 5% blasts in the bone marrow as assessed by morphology or flow cytometry; however, if an adequate bone marrow sample cannot be obtained, patients may be enrolled if there is unequivocal evidence of leukemia with >= 5% blasts in the peripheral blood; in addition, patients in all categories must not be eligible to undergo curative therapy, such as immediate stem cell transplantation (SCT) Diagnosis of acute leukemia Patient has newly diagnosed acute lymphoblastic leukemia (ALL) or acute undifferentiated leukemia with >= 25% blasts in the bone marrow (M3), with or without extramedullary disease; patients with T-cell ALL are eligible; patients with bilineage or biphenotypic acute leukemia are eligible, provided the morphology and immunophenotype are predominantly lymphoid leukemia Patients who are considered appropriate candidates for allogeneic hematopoietic stem cell transplantation and have one of the following diagnoses:\r\n* Acute lymphocytic leukemia in first or subsequent remission\r\n* Acute myeloid leukemia in first or subsequent remission \r\n* Acute lymphocytic leukemia in relapse or primary refractory disease with a circulating blast count of no more than 10,000/mm^3 (Arms A or C only)\r\n* Acute myeloid leukemia in relapse or primary refractory disease with a circulating blast count of no more than 10,000/mm^3 (Arms A or C only)\r\n* Refractory anemia with excess blasts (RAEB-1 and RAEB-2) (if the patient has received previous induction chemotherapy within 60 days)\r\n* Chronic myelogenous leukemia with a history of accelerated phase or blast crisis (if the patient has received at least one course of induction chemotherapy)\r\n* Other acute leukemia or related neoplasm (including but not limited to ‘biphenotypic’, ‘undifferentiated’ or ‘ambiguous lineage’ acute leukemia, blastic plasmacytoid dendritic cell neoplasm or lymphoblastic lymphoma) Acute myeloid leukemia, including biphenotypic acute leukemia or mixed-lineage leukemia:\r\n* All patients must be in CR as defined by hematologic recovery and < 5% blasts by morphology/flow cytometry in a representative bone marrow sample with cellularity >= 15% for age; patients who do not have high-risk features (for example preceding myelodysplastic syndrome [MDS], high-risk cytogenetics, >= 2 cycles to obtain CR, erythroblastic or megakaryocytic leukemia or >= CR2) must be discussed with the principal investigator (PI) prior to enrollment and at the Patient Care Conference or equivalent group such as the pediatric leukemia board as an alternative\r\n* Patients in whom adequate marrow/biopsy specimens cannot be obtained to determine remission status by morphologic assessment, but have fulfilled criteria of remission by flow cytometry, recovery of peripheral blood counts with no circulating blasts, and/or normal cytogenetics (if applicable) may still be eligible; reasonable attempts must be made to obtain an adequate specimen for morphologic assessment, including possible repeat procedures; these patients must be discussed with the PI prior to enrollment; patients persistently aplastic for greater than one month since completing last chemotherapy are also eligible with PI approval Acute lymphocytic leukemia (ALL) – must have < 5% marrow blasts at the time of transplant Refractory acute leukemia (> 5% blasts) or progressive disease Have acute leukemia. Acute biphenotypic or bilineal leukemia in first or greater complete remission Diagnosis of acute leukemia by World Health Organization (WHO) criteria (e.g.-acute myeloid leukemia, acute lymphoblastic leukemia, acute leukemia of ambiguous origin) Histologically confirmed diagnosis of relapsed or refractory acute lymphoblastic leukemia (ALL) (including Burkitt leukemia), acute myeloid leukemia (AML), mixed lineage leukemia, biphenotypic leukemia, or chronic myelogenous leukemia (CML) in blast crisis\r\n* Refractory disease defined as: persistent disease after at least two induction cycles\r\n* Relapsed disease: second or subsequent relapse, any relapse refractory to salvage chemotherapy Acute leukemia not in remission Newly diagnosed or relapsed mixed phenotype acute leukemia (MPAL), which for this protocol, will be defined as follows: bone marrow result interpreted by the reading pathologist (or tissue biopsy for cases of extramedullary disease) as: biphenotypic leukemia, bilineal leukemia, undifferentiated leukemia, mixed lineage leukemia, leukemia of ambiguous lineage, T/myeloid leukemia, B/myeloid leukemia, or other diagnosis indicating the presence of multiple lineages within the cell population Patients with biopsy-proven acute lymphoblastic leukemia, acute myeloid leukemia, or myelodysplastic syndrome in remission or relapse Relapsed or refractory acute leukemia (acute myeloid leukemia or acute lymphoblastic leukemia or lymphoma) in second or subsequent remission, with remission defined as < 5% bone marrow blasts morphologically Acute myeloid leukemia/acute lymphoblastic leukemia, including biphenotypic acute leukemia or mixed-lineage leukemia: Must have < 5% morphologic marrow blasts in an evaluable marrow sample (> 25% of normal cellularity for age collected less than one month prior to start of conditioning; patients in which adequate marrow/biopsy specimens cannot be obtained to determine remission status by morphologic assessment, but have fulfilled criteria of remission by flow cytometry, recovery of peripheral blood counts with no circulating blasts, and/or normal cytogenetics (if applicable) may still be eligible; reasonable attempts must be made to obtain an adequate specimen for morphologic assessment, including possible repeat procedures; these patients must be discussed with the principal investigator prior to enrollment; patients persistently aplastic for greater than one month since completing last chemotherapy are also eligible with principal investigator (PI) approval Acute myeloid leukemia/acute lymphoblastic leukemia, including biphenotypic acute leukemia or mixed-lineage leukemia: Must have < 5% morphologic marrow blasts in an evaluable marrow (> 25% of normal cellularity for age) collected less than one month prior to start of conditioning; patients persistently aplastic for greater than one month since completing last chemotherapy are also eligible with the approval of the PI or designee Diagnosis of acute lymphoblastic leukemia (ALL) or biphenotypic leukemia meeting the following criteria: Based on European Group for the Classification of Acute Leukemia (EGIL) diagnostic criteria Must not have acute leukemia Diagnosis of refractory or relapsed ALL (acute lymphocytic leukemia) Malignancies included are:\r\n* Acute leukemia, including Acute myeloid leukemia (AML), biphenotypic acute leukemia or mixed-lineage leukemia, acute lymphoblastic leukemia (ALL); all patients must be in complete response (CR) as defined by < 5% blasts by morphology/flow cytometry in a representative bone marrow sample with adequate cellularity to assess remission status\r\n* Myelodysplasia (MDS) International Prognostic Scoring System (IPSS) intermediate (Int)-2 or high risk (i.e., RAEB, RAEBt) or refractory anemia with severe pancytopenia or high risk cytogenetics; blasts must be < 10% by a representative bone marrow aspirate morphology\r\n* Chronic Myeloid Leukemia excluding refractory blast crisis; to be eligible in first chronic phase (CP1) patient must have failed or be intolerant to tyrosine kinase inhibitor therapy Patients with a diagnosis of relapsed or refractory hematologic malignancy including, but not limited to acute myeloid leukemia (AML), acute lymphoblastic leukemia (ALL), Burkitt’s leukemia/lymphoma, prolymphocytic leukemia, biphenotypic acute leukemia, blast-phase of chronic myeloid leukemia (CML), B-cell lymphoma, or Richter’s transformation of chronic lymphocytic leukemia (CLL) Participants whose leukemia meets WHO diagnostic criteria of mixed phenotype acute leukemia (MPAL) or leukemia of ambiguous lineage are not eligible Bi-phenotypic acute leukemia Acute leukemia of ambiguous lineage (biphenotypic leukemia) Patients with any of the following oncologic diagnoses are not eligible:\r\n* Any concurrent malignancy\r\n* Juvenile myelomonocytic leukemia (JMML)\r\n* Philadelphia chromosome positive AML\r\n* Biphenotypic or bilineal acute leukemia\r\n* Acute promyelocytic leukemia\r\n* Acute myeloid leukemia arising from myelodysplasia\r\n* Therapy-related myeloid neoplasms\r\nNote: enrollment may occur pending results of clinically indicated studies to exclude these conditions Patients with one of the following diagnoses:\r\n* Acute myeloid leukemia (AML)\r\n* Acute lymphoblastic leukemia (ALL)\r\n* Juvenile myelomonocytic leukemia (JMML)\r\n* Myelodysplastic syndrome (MDS) Patients who are considered appropriate candidates for allogeneic hematopoietic stem cell transplantation and have one of the following diagnoses:\r\n* Acute lymphocytic leukemia in first or subsequent remission\r\n* Acute myeloid leukemia in first or subsequent remission \r\n* Acute lymphocytic leukemia in relapse or primary refractory disease with a circulating blast count of no more than 10,000/mm^3\r\n* Acute myeloid leukemia in relapse or primary refractory disease with a circulating blast count of no more than 10,000/mm^3 \r\n* Refractory anemia with excess blasts (RAEB-1 and RAEB-2) \r\n* Chronic myelogenous leukemia with a history of accelerated phase or blast crisis\r\n* Other acute leukemia (including but not limited to ‘biphenotypic’, ‘undifferentiated’ or ‘ambiguous lineage’ acute leukemia) AML or acute leukemia of ambiguous lineage:\r\n* If relapse AML or acute leukemia of ambiguous lineage:\r\n** Must have a prior diagnosis of AML or acute leukemia of ambiguous lineage and be in 1st or greater relapse as evidenced by morphology, immunophenotype, molecular or cytogenetic signature\r\n** Must not have received prior reinduction therapy for this relapse\r\n* If primary refractory AML or acute leukemia of ambiguous lineage:\r\n** Must have had a prior diagnosis of AML or acute leukemia of ambiguous lineage and\r\n** Must not have received more than 3 previous induction attempts\r\n* If primary AML or acute leukemia of ambiguous lineage newly diagnosed or in remission\r\n** Primary treating oncologist must have deemed patient unable to complete standard treatment therapy and selected FLAG as appropriate alternative regimen\r\n* If prior diagnosis of acute lymphoblastic leukemia must have evidence of transformation to AML or acute leukemia of ambiguous lineage as evidenced by morphology, immunophenotype, molecular or cytogenetic signature\r\n* Patients meeting above criteria are eligible regardless of central nervous system (CNS) classification Disease associated with secondary MF such as metastatic carcinoma, lymphoma, myelodysplasia, hairy cell leukemia, mast cell disease or acute leukemia (including M7 disease or acute panmyelosis with MF) Acute myeloid leukemia, including biphenotypic acute leukemia or mixed-lineage leukemia\r\n* All patients must have acute myeloid leukemia (AML) that is considered best treated by stem cell transplant by the referring physician and the attending transplant physician\r\n* All patients must be in complete remission (CR) as defined by < 5% blasts by morphology/flow cytometry in a representative bone marrow sample with cellularity >= 15% for age\r\n* Patients in which adequate marrow/biopsy specimens cannot be obtained to determine remission status by morphologic assessment, but have fulfilled criteria of remission by flow cytometry, recovery of peripheral blood counts with no circulating blasts, and/or normal cytogenetics (if applicable) may still be eligible; reasonable attempts must be made to obtain an adequate specimen for morphologic assessment, including possible repeat procedures; these patients must be discussed with the principal investigator prior to enrollment Acute myelogenous leukemia diagnosed by World Health Organization (WHO) criteria with one of the following:\r\n* Primary refractory disease following >= 1 cycle of induction chemotherapy\r\n* First relapse or higher; patients with primary or secondary acute myelogenous leukemia are eligible; patients with biphenotypic leukemia are eligible FAB subtype M0 (minimally differentiated acute myeloblastic leukemia), M6 (acute erythroid leukemias, including erythroleukemia (M6a) and pure erythroid leukemia (M6b)), or M7 (acute megakaryoblastic leukemia) Have an acute leukemia Bi-phenotypic acute leukemia Biphenotypic leukemia Biphenotypic leukemia Acute lymphocytic leukemia (ALL) must have < 5% marrow blasts at the time of HCT Recipients with primary or secondary acute leukemia, refractory anemia with excess blasts (RAEB), CML, or other eligible diagnosis in transformation to acute leukemia must have =< 5% blasts in bone marrow and no circulating blasts in peripheral blood at study entry; recipients who do not meet these criteria may be re-evaluated for study eligibility after receiving standard induction therapy for acute leukemia and determined to be in remission Burkitt’s or mixed lineage leukemia, T cell ALL Prior morphological diagnosis of AML other than acute promyelocytic leukemia according to the 2001 World Health Organization (WHO) diagnostic criteria; patients with biphenotypic, RAS-mutated acute leukemia are also eligible The patient must have discontinued all previous therapies for acute leukemia for at least 14 days and recovered from the acute non-hematologic side effects of the therapy T cell prolymphocytic leukemia Have current acute or chronic leukemia. Participant with acute leukemia must meet one of the following criteria:\r\n* Relapsing acute lymphoblastic leukemia (ALL), acute myeloid leukemia (AML), or acute biphenotypic leukemia in 2nd or greater relapse, or\r\n* Refractory ALL, AML, or acute biphenotypic leukemia failing to achieve CR with >= 2 induction or re-induction attempts Must not have leukemia. Acute myeloid leukemia, myelodysplasia, acute lymphoblastic leukemia, chronic myeloid leukemia, or myeloproliferative neoplasms requiring a matched allogeneic hematopoietic stem cell transplant (HSCT)\r\n* Acute leukemia (acute myeloid leukemia [AML] or acute lymphoblastic leukemia [ALL]) must be in complete remission defined as: < 5% marrow blasts with no morphologic evidence of leukemia, no peripheral blasts, marrow > 20% cellular, and peripheral absolute neutrophil count > 1000/uL (platelet recovery is not required)\r\n* Myelodysplasia (MDS) and chronic myeloid leukemia (CML): must have < 5% marrow blasts\r\n* Myeloproliferative neoplasms (MPN): must have < 5% peripheral/marrow blasts Acute or chronic leukemia. Have current acute or chronic leukemia Have a diagnosis of lung cancer (both small cell and non-small cell), central nervous system tumors, acute myeloid leukemia, and acute lymphoblastic leukemia will be excluded Acute myelogenous leukemia diagnosed by World Health Organization (WHO) criteria with one of the following (patients with biphenotypic leukemia are eligible, provided that the treating physician determines an acute myeloid leukemia (AML) treatment regimen is appropriate)\r\n* Primary refractory disease following >= 1 cycle of induction chemotherapy\r\n* First relapse or higher; patients with primary or secondary acute myelogenous leukemia are eligible Acute or chronic leukemia diagnosis Those with acute leukemia must be in remission at the time of transplant Any patient with non-acute promyelocytic leukemia (APL) acute leukemia (acute myeloid leukemia [AML], acute lymphoblastic leukemia [ALL], biphenotypic leukemia) undergoing curative intent chemotherapy OR any patient undergoing allogeneic hematopoietic stem cell transplantation (HSCT) for a hematologic disorder (including acute leukemia as above, chronic myelogenous leukemia [CML], chronic lymphocytic leukemia [CLL], myelodysplastic syndrome [MDS], primary or secondary myelofibrosis, hypereosinophilic syndromes, plasma cell disorders, B-cell or T-cell lymphoma) Patients with acute myelogenous leukemia (AML) or relapsed acute lymphoblastic leukemia (ALL) that will receive chemotherapy with/without transplant must have, or be scheduled to have, an external tunneled CVC (Broviacs, Hickmans, tunneled PICCs, etc.) and/or non-tunneled PICC that is expected to remain in place for an additional >= 3 months Patients with a hematologic malignancy other than aplastic anemia or primary myelofibrosis, scheduled to undergo RIC allogeneic SCT with a peripheral blood stem cell graft from an unrelated donor, using fludarabine phosphate (Flu)/busulfan (Bu) conditioning and tacrolimus (Tac)/methotrexate (MTX) GVHD prophylaxis; the following diagnoses are included:\r\n* Acute leukemia - acute myeloid leukemia (AML), acute lymphoblastic leukemia (ALL) or acute biphenotypic leukemia; patients will have documentation of complete remission within 6 weeks prior to their transplant; complete remission is defined as < 5% blasts on a bone marrow biopsy and absence of any known extramedullary disease\r\n* Chronic myelogenous leukemia in any stage, but with documentation of < 5% blasts on a bone marrow biopsy within 6 weeks prior to transplant\r\n* Myelodysplastic syndrome of any subtype, but with documentation of < 5% blasts on a bone marrow biopsy within 6 weeks prior to transplant\r\n* Myeloproliferative disorders other than primary myelofibrosis\r\n* Lymphoma - all types of lymphoma are eligible\r\n* Chronic lymphocytic leukemia (CLL) and prolymphocytic leukemia (PLL) Acute leukemia receiving induction chemotherapy All acute myelogenous leukemia and high-risk myelodysplastic leukemia patients who are admitted to the leukemia service and those who are referred from other services (i.e. pediatrics, medical oncology, etc.) will be eligible for the study Intensive induction chemotherapy for a new diagnosis of acute myelogenous leukemia, acute lymphocytic leukemia, or myelodysplastic syndrome receiving standard anthracycline based chemotherapy New diagnosis of de novo ALL, lymphomas (T or B cell), or mixed-phenotype acute leukemia Diagnosis of acute myeloid leukemia (AML), acute biphenotypic leukemia, or acute lymphoblastic leukemia (ALL); CML transformed to blast crisis is eligible Acute leukemia (acute myeloid leukemia [AML] or acute lymphoblastic leukemia [ALL]) must be in complete remission defined as: < 5% marrow blasts with no morphologic evidence of leukemia, no peripheral blasts, marrow > 20% cellular, and peripheral absolute neutrophil count > 1000/uL (platelet recovery is not required) Acute lymphocytic leukemia (ALL) – must have < 5% marrow blasts at the time of transplant Acute undifferentiated leukemia (AUL) Acute biphenotypic leukemia COHORT 1: RELAPSED/REFRACTORY LEUKEMIA \r\n* Acute lymphoblastic leukemia, first or greater relapse\r\n* Acute myeloid leukemia, first or greater relapse\r\n* Leukemia refractory to induction chemotherapy\r\n* Other recurrent leukemia\r\n* Myelodysplastic syndrome (MDS), first or greater relapse, or refractory to initial therapy COHORT 2: NEW DIAGNOSIS\r\n* Acute myeloid leukemia, new diagnosis\r\n* New diagnosis infant mixed-lineage leukemia (MLL)-rearranged acute lymphoblastic leukemia (ALL) or low hypodiploid (< 40 chromosomes) ALL\r\n* Rare leukemia- e.g., juvenile myelomonocytic leukemia (JMML), leukemia of ambiguous lineage\r\n* Secondary leukemia\r\n* Myelodysplastic syndrome (MDS) not eligible for stem cell transplant