Patients must have newly diagnosed B lymphoblastic leukemia (2008 World Health Organization [WHO] classification) (also termed B-precursor acute lymphoblastic leukemia); patients with Down syndrome are also eligible Patients with T-lymphoblastic leukemia (T-ALL)/lymphoblastic lymphoma (T-LL) are not eligible Patients with B-lymphoblastic lymphoma (B-LL) are not eligible Patients must have newly diagnosed B lymphoblastic leukemia (2008 World Health Organization [WHO] classification) (also termed B-precursor acute lymphoblastic leukemia) or acute leukemia of ambiguous lineage (ALUL), which includes mixed phenotype acute leukemia (MPAL); for patients with ALUL, the morphology and immunophenotype must be at least 50% B lymphoblastic Patients with secondary B acute lymphoblastic leukemia (B-ALL) that developed after treatment of a prior malignancy with cytotoxic chemotherapy Newly diagnosed patients with CD-22 positive B-cell acute lymphoblastic leukemia (World Health Organization [WHO] criteria) are eligible; patients with Burkitt type ALL are NOT eligible Patients with previously untreated acute lymphoblastic leukemia (B-cell or T-cell) Philadelphia (Ph)-positive ALL, Burkitt’s leukemia/lymphoma, or lymphoblastic lymphoma Diagnosis of acute lymphoblastic leukemia or lymphoblastic lymphoma as defined by the World Health Organization Total bilirubin < 2 mg/dL (unless due to acute lymphoblastic leukemia [ALL]) Acute lymphoblastic leukemia (ALL): Lymphoblastic lymphoma Diagnosis of T-acute lymphoblastic leukemia/ lymphoblastic lymphoma according to World Health Organization (WHO) criteria which has relapsed or is refractory to chemotherapy High grade lymphomas (Burkitts or lymphoblastic), plasma cell leukemia. Acute lymphoblastic leukemia, excluding known mature B-cell ALL by the presence of any of the following: surface immunoglobulin, L3 morphology, t(8;14)(q24;q32), t(8;22), or t(2;8) Patients with lymphoblastic lymphoma are also eligible Group A: CD19+ B-acute lymphoblastic leukemia (ALL) undergoing allogeneic HSCT or Must have relapsed or refractory precursor B-cell acute lymphoblastic leukemia or acute lymphoblastic lymphoma Patients with first relapsed/refractory B-cell acute lymphoblastic leukemia (ALL) Relapsed or refractory CD19-positive B-lineage acute lymphoblastic leukemia having received at least 1 prior line of therapy Relapsed and/or refractory Philadelphia negative acute lymphoblastic leukemia or lymphoblastic lymphoma (Lead-in and Phase 2) Relapsed and/or refractory Philadelphia positive acute lymphoblastic leukemia, Burkitt leukemia/lymphoma or “double-hit” leukemia/lymphoma (2 separate cohorts, phase II only) PHASE I: Diagnosis of CD22-positive acute lymphoblastic leukemia PHASE II: Diagnosis of CD22-positive acute lymphoblastic leukemia Patients must have a confirmed diagnosis of either:\r\n* Acute lymphoblastic leukemia\r\n* Lymphoblastic lymphoma with detectable abnormal blasts in the bone marrow Diagnosis of T acute lymphoblastic leukemia (T-ALL) or Burkitt’s leukemia/lymphoma Patient with relapsed or refractory CD19 positive B-acute lymphoblastic leukaemia (B-ALL) Patients with CLL, Burkitt or lymphoblastic lymphoma are excluded Burkitt’s leukemia or lymphoma, T-cell ALL or lymphoblastic lymphoma WHO classification defined B-lymphoid malignancy, with the exception of Burkitt lymphoma/leukemia, plasma cell myeloma, acute lymphoblastic leukemia, lymphoblastic lymphoma, and plasmablastic lymphoma. Newly diagnosed acute lymphoblastic leukemia/lymphoma Research participants with precursor B-cell acute lymphoblastic leukemia/lymphoma or plasma cell dyscrasias Diagnosis of acute lymphoblastic leukemia or lymphoblastic lymphoma Patients will be excluded if they have isolated extra-medullary relapse of acute lymphoblastic leukemia (ALL) Ph-positive ALL, Burkitt’s leukemia or lymphoma, T-cell ALL or lymphoblastic lymphoma Patients with acute lymphoblastic leukemia (ALL) or high grade (stage III or IV) non-Hodgkin lymphoma (NHL) after first relapse or with primary refractory disease Documented acute lymphoblastic leukemia (ALL) or lymphoblastic lymphoma (LL) as defined by the criteria below: Patients must have a confirmed diagnosis of either B- or T-cell acute lymphoblastic leukemia or lymphoblastic lymphoma that is either:\r\n* Arm A: Initially diagnosed at age 40 or later, OR\r\n* Arm B: Relapsed after or failed to respond to >= 1 previous chemotherapy regimen Patients must have B-ALL, or previously diagnosed B lymphoblastic lymphoma (B-LL), with >= 5% (M2 or M3) bone marrow blasts with or without extramedullary disease\r\n* NOTE: Relapsed patients previously diagnosed with B-lymphoblastic lymphoma (B-LL) are eligible if they have an M2 or M3 marrow at the time of enrollment on this study Confirmed diagnosis of acute lymphoblastic leukemia or lymphoblastic lymphoma Have acute T-cell lymphoblastic leukemia (T-ALL) or T-cell lymphoblastic lymphoma (T-LBL). STRATUM III: \r\n* Any patients with MDD >= 1% and MRD positive (>= 0.01%) on day 8 in T-lymphoblastic lymphoma EXCLUDING: Burkitt's lymphoma and precursor B lymphoblastic leukemia/lymphoma Acute Leukemias or T lymphoblastic lymphoma in 1st or subsequent complete remission (CR): Acute lymphoblastic leukemia (ALL)/T lymphoblastic lymphoma; acute myelogenous leukemia (AML); acute biphenotypic leukemia (ABL); acute undifferentiated leukemia (AUL) Patients of all ages with newly diagnosed, previously untreated CD-20+ acute lymphoblastic leukemia (ALL), or lymphoblastic lymphoma, Burkitt leukemia/lymphoma or having achieved complete remission (CR) with one course of induction chemotherapy Previously untreated Philadelphia chromosome negative acute lymphoblastic leukemia/lymphoma Participants must have relapsed or refractory acute lymphoblastic leukemia or lymphoma:\r\n* Stratum I: T-cell lymphoblastic leukemia or lymphoma in first relapse or refractory to one or two courses of frontline induction therapy\r\n* Stratum II: B-cell or T-cell lymphoblastic leukemia or lymphoma in second or third relapse or refractory to 2 or 3 induction or re-induction attempts; patients with Philadelphia chromosome positive (Ph+) acute lymphoblastic leukemia (ALL) must be refractory or relapsed after treatment with a regimen that included a tyrosine kinase inhibitor (TKI) Patients must have a diagnosis of acute lymphoblastic leukemia (ALL) with ? 25% blasts in the bone marrow (M3), with or without extramedullary disease. Patients must have precursor-B lymphoblastic leukemia or lymphoma Acute Lymphoblastic Leukemia (ALL) in CR NK cell lymphoblastic leukemia in any CR Pathologically confirmed diagnosis of B-lineage acute lymphoblastic leukemia, Burkitt leukemia or lymphoma, or B-lineage lymphoblastic lymphoma Survivor of childhood acute lymphoblastic leukemia (ALL) or non-Hodgkins's lymphoma (NHL) (treated for ALL or NHL before the age of 21 years old and ?5 years post-treatment) Burkitt and/or precursor lymphoblastic leukemia/lymphoma Acute lymphoblastic leukemia (ALL) in 1st or subsequent remission Subjects with T-cell acute lymphoblastic leukemia or T-cell lymphoblastic lymphoma refractory to or relapsed from standard therapies ALL or lymphoblastic lymphoma patients in first or higher relapse Patients with biopsy-proven acute lymphoblastic leukemia, acute lymphoblastic lymphoma, or acute biphenotypic leukemia in remission or relapse PARENT/CAREGIVER: Adult primary caregiver of children treated for acute lymphoblastic leukemia (ALL) or acute myeloid leukemia (AML) or lymphoblastic lymphoma (LL) and daily contact with the child Currently being treated in the maintenance phase of therapy for pediatric ALL or lymphoblastic lymphoma Active relapse of ALL or lymphoblastic lymphoma Currently enrolled on any therapeutic research study for the treatment of ALL or lymphoblastic lymphoma Currently enrolled and receiving treatment for acute lymphoblastic leukemia (ALL) on the TOTXVI therapy protocol at St. Jude Children’s Research Hospital Patients with:\r\n* Acute lymphoblastic leukemia/lymphoma \r\n* Malignant brain tumor \r\n* Non-central nervous system (CNS) solid tumors \r\n* Acute myeloblastic leukemia\r\n* Non-Hodgkin’s lymphoma \r\n* Hodgkin’s disease\r\n* Head and neck tumors Acute lymphoblastic leukemia, including T lymphoblastic lymphoma with a history of marrow involvement Acute lymphoblastic leukemia (ALL) in first or higher remission (< 5% blasts in the bone marrow) Clinical Diagnosis of one of the following: Acute Lymphoblastic Leukemia, Acute Myelogenous Leukemia, Lymphoma Acute lymphoblastic leukemia (ALL)