DISEASE CHARACTERISTICS:

• Diagnosis of 1 of the following hematologic malignancies:

  • Acute myeloid leukemia in complete remission (CR), as defined by hematologic recovery and < 5% blasts by morphology within the bone marrow and a cellularity of ≥ 15%

    • Meets 1 of the following criteria:

      • In first complete remission (CR1) with any of the following high-risk features:

        • Preceding myelodysplastic syndromes (MDS)
        • High-risk cytogenetics (e.g., monosomy 5 or 7)
        • At least 2 courses of therapy were required to obtain CR
        • Erythroblastic or megakaryocytic leukemia
      • In ≥ second CR (CR2)

    • No acute leukemia in relapse, as defined by ≥ 5% marrow blasts by morphology

      • Patients < 21 years of age in early relapse, as defined by presence of minimal residual disease but in morphologic remission, are eligible

  • Acute lymphoblastic leukemia in CR, as defined by hematologic recovery and < 5% blasts by morphology within the bone marrow and a cellularity of ≥ 15%

    • Meets 1 of the following criteria:

      • In CR1 with any of the following high-risk features:

        • t[9;22], t[1;19], t[4;11], or other MLL rearrangements
        • Hypodiploid
        • More than 1 course of therapy was required to obtain CR
      • In ≥ CR2
    • No acute leukemia in relapse, as defined by ≥ 5% marrow blasts by morphology

  • Chronic myelogenous leukemia

    • Patients in blast crisis (BC) are eligible provided they received therapy and achieved accelerated or chronic phase

      • Patients who remain in BC are not eligible
    • Patients in first chronic phase are eligible provided they have failed or are intolerant to imatinib mesylate
  • Advanced myelofibrosis

  • MDS

    • Meets 1 of the following criteria:

      • International Prognostic Scoring System (IPSS) score intermediate-2 or high risk (refractory anemia [RA] with excess blasts [RAEB] or RAEB in transformation)
      • RA with severe pancytopenia
      • High-risk cytogenetics
    • Blasts must be < 10% morphologically in representative bone marrow aspirate (obtained < 2 weeks prior to study enrollment)

  • Lymphoblastic lymphoma, Burkitt's lymphoma, or other high-grade non-Hodgkin's lymphoma (NHL)

    • Meets 1 of the following stage criteria:

      • Stage I or II disease (for < 1 year) that progressed
      • Stage III or IV disease that was previously treated with initial therapy and subsequently progressed after being in CR or partial remission (PR)
    • No chemotherapy-refractory disease, defined as progressive disease after > 2 salvage regimens

  • Chronic lymphocytic leukemia/small lymphocytic lymphoma, marginal zone B-cell lymphoma, lymphoplasmacytic lymphoma, or follicular lymphoma

    • Progressed after at least two different prior therapies
    • Patients with bulky disease (nodal mass > 5 cm) should be considered for debulking chemotherapy before transplantation

  • Mantle cell lymphoma or prolymphocytic leukemia

    • Received initial therapy
    • In ≥ CR1 or ≥ first PR (PR1)

  • Large cell NHL

    • In > CR2 or > second PR (PR2)

      • Patients in CR2 or PR2 with initial short remission (< 6 months) are eligible
    • No chemotherapy-refractory disease, defined as progressive disease after > 2 salvage regimens

  • Multiple myeloma

    • In > PR2
    • Patients with chromosome 13 abnormalities, first response lasting < 6 months, or β-2 microglobulin > 3 mg/L, may be eligible provided they have received initial therapy
• No active CNS leukemia involvement, defined as cerebrospinal fluid with > 5 WBC/mm^3 and malignant cells on cytospin
• No 5/6 or 6/6 HLA-matched related donor available
• No existing 0-1 HLA-A, -B, -C, -DRB1 and -DQB1 matched related or unrelated donor available

• Umbilical cord blood (UCB) units available

  • Two UCB units must be 0-2 HLA-A, -B, and - DRB1 mismatched with the recipient and to each other

    • One unit ≥ 2.5 X 10^7 total nucleated cell (TNC)/kg (for unmanipulated cells) AND one unit ≥ 1.5 X 10^7 TNC/kg (for ex- vivo expansion cells)
  • A third unit of UCB ≥ 1.5 X 10^7 TNC/kg AND 0-2 HLA-A, -B, and -DRB1 mismatched with the recipient and to the other two UCB units NOTE: A new classification scheme for adult non-Hodgkin's lymphoma has been adopted by PDQ. The terminology of "indolent" or "aggressive" lymphoma will replace the former terminology of "low", "intermediate", or "high" grade lymphoma.

However, this protocol uses the former terminology.

PATIENT CHARACTERISTICS:

• Creatinine ≤ 2.0 mg/dL (adults)
• Creatinine clearance > 60 mL/min (pediatrics)
• Bilirubin ≤ 2 times upper limit of normal (ULN)
• AST/ALT ≤ 2 times ULN
• Corrected DLCO > 50% normal
• LVEF ≥ 45%
• Karnofsky performance status 70-100% (adults) OR Lansky performance status 50-100% (pediatrics)
• Not pregnant or nursing
• Negative pregnancy test
• Fertile patients must use effective double-method contraception
• No uncontrolled viral or bacterial infection
• No active or recent (within the past 6 months) invasive fungal infection without Infectious Disease consult and approval
• No HIV seropositivity

PRIOR CONCURRENT THERAPY:

• See Disease Characteristics

• No prior autologous or allogeneic stem cell transplantation with a myeloablative preparative regimen

  • If ≤ 18 years of age, prior myeloablative transplantation allowed provided it was completed > 6 months ago
• More than 28 days since prior experimental drugs
• No other concurrent experimental drugs