Inclusion Criteria:

          -  Eligible Diseases

               1. Non-Hodgkin's Lymphoma (NHL)

                    -  Patients with chemo-sensitive histologically confirmed NHL will be eligible
                       for this treatment protocol contingent on histologic sub-classification.

                    -  Patients in partial or complete remission following cell therapy will also
                       be eligible.

                    -  NHL patients with resistant or refractory lymphoma (no PR following up to
                       three cycles of combination chemotherapy) will not be eligible for
                       transplant in this trial.

                    -  Lymphoblastic Lymphoma:

                         1. All patients will be eligible in second or greater complete remission
                            (CR) or first or subsequent partial remission (PR)

                         2. Patients with any high-risk features will be eligible in first complete
                            remission

                         3. High risk features include: Stage IV, LDH >2 x upper limit of normal, ≥
                            2 extranodal sites

                    -  Mature B-cell Lymphoma

                         1. Follicular Lymphoma and other indolent lymphoma in ≥ second CR2/PR2

                         2. Diffuse Large B-Cell Lymphoma: in ≥ CR2 or ≥ PR1; a high intermediate
                            or high IPI (≥ 2 for age-adjusted IPI or ≥3 for IPI) at diagnosis and
                            double-hit or triple-hit lymphoma will be eligible in first CR;
                            transformed lymphoma from FL (or other indolent lymphoma) or chronic
                            lymphocytic leukemia will be eligible if chemosensitive and bone marrow
                            is negative

                         3. Mantle Cell Lymphoma: in first or greater CR or PR

                         4. Burkitt's/Burkitt's like: all patients except localized lymphoma will
                            be eligible any time after initial therapy (after achievement of first
                            complete remission), or in partial remission if they fail to achieve
                            CR; patients with localized (stage I or Ziegler stage A) will be
                            eligible only if they fail to achieve CR1 or after relapse

                    -  Mature T-Cell Lymphoma

                         1. Chemosensitive T-cell lymphomas including Primary T-cell not otherwise
                            specified angioimmunoblastic, and ALK-positive anaplastic large cell,
                            will be eligible after initial therapy, whether or not CR is achieved.

                         2. Mycosis fungoides/Sezary syndrome will be eligible in ≥CR2/PR2

               2. Hodgkin Lymphoma (HL)

                    -  Patients with histologically proven HL will be eligible for transplantation
                       after failing prior therapy.

                    -  Patients with resistant disease (initial or at relapse): those who fail to
                       achieve an objective partial response to three cycles of combination
                       non-cross resistant chemotherapy will not be eligible for transplant in this
                       trial.

                    -  For stage I/II patients treated with primary chemotherapy-radiation, they
                       must have failed (no CR or progression after CR) at least one salvage
                       combination chemotherapy treatment regimen

                    -  For advanced (stage III/IV) Hodgkin disease, patients must have failed an
                       Adriamycin containing regimen (ABVD) or an alternative non-cross resistant
                       regimen (e.g. MOPP)

                    -  Patients with any high-risk features will also be eligible, including those
                       who:

                         1. fail to achieve complete remission with initial combination
                            chemotherapy

                         2. have bulky disease after initial therapy (chemotherapy or radiation)
                            defined as residual mediastinal mass ≥ 5 cm or other residual mass ≥ 10
                            cm accompanied by other features of persisting disease (e.g., PET scan
                            positive; high LDH; enlarging on serial x-rays or biopsy positive) will
                            be eligible - if feasible, persistent disease should be proven by
                            biopsy

                    -  Patients should receive chemotherapy to attempt to achieve CR or minimal
                       disease state for all patients pre-transplantation. The use of up to three
                       cycles of non-cross resistant combination chemotherapy is advised.

                    -  Residual areas of limited disease should be considered for radiotherapy
                       after and not prior to transplantation.

               3. HIV positive patients who are otherwise eligible for this study may be enrolled
                  if they meet the following requirements:

                    -  Are seen in the infectious disease (ID)/HIV clinic prior to enrollment on
                       study for the purpose of determining eligibility and for local coordination
                       of HIV care during the peri-transplant period.

                    -  Are on maximally active anti-HIV regimen to control disease as determined
                       appropriate by the ID/HIV physicians. For the majority of patients, this
                       will be a highly active anti-retroviral therapy (HAART)-type therapy
                       including a protease inhibitor.

                    -  CD4+ ≥ 50/µL

                    -  HIV RNA viral load ≤ 100,000 copies per mL on each of samples 4 weeks apart.
                       The most recent level must be within 30 days of enrollment.

          -  Performance Status: Karnofsky Performance Status ≥ 80% for patients ≥ 16 years of age
             or Lansky Play Score ≥ 80 for patients < 16 years of age. Note: if poor performance
             status is due to lymphoma - KPS ≥ 60% or LPS ≥ 60 is acceptable

          -  Organ Function

             1. No evidence of serious organ dysfunction that is not attributable to tumor
             including:

               1. Hematologic:

                    -  hemoglobin > 8 gm/dL

                    -  WBC > 2.5 x 109/L with an ANC > 1.5 x 109/L off G-CSF or GM-CSF for 10 days
                       or Neulasta for 21 days

                    -  platelets > 100 x 109/L without transfusion

                    -  bone marrow cellularity of > 20% with <5% involvement with tumor

               2. Renal: GFR > 50 ml/min/1.73m2 or serum creatinine ≤ 2.5 x ULN for age

               3. Hepatic: no history of severe prior or ongoing chronic liver disease. Total
                  bilirubin ≤ 2.0 mg/dl, AST and alkaline phosphatase <5x upper limit of normal

               4. Cardiac: free of symptoms of uncontrolled cardiac disease including unstable
                  angina, decompensated congestive heart failure, or arrhythmia. The ejection
                  fraction by gated cardiac blood flow scan (MUGA) or Echocardiogram must be >40%

               5. Pulmonary: no significant obstructive airways disease (FEV1 must be ≥ 50%) and
                  must have acceptable diffusion capacity (corrected DLCO > 50% of predicted)

               6. Central Nervous System: Patients with a history of CNS involvement by lymphoma or
                  with relapsed primary CNS lymphoma will be eligible for Cy/TBI arm. Patients with
                  active CNS disease are eligible if they have completed a standard treatment for
                  CNS lymphoma and have no evidence of progressive CNS disease at the time of
                  enrollment

          -  Other Inclusion Criteria

               1. At least 4 weeks from previous chemotherapy; 6 weeks from nitrosoureas

               2. Women of child bearing potential and sexually active males with partners of child
                  bearing potential must agree to use adequate birth control for the duration of
                  treatment

               3. Patients who are carriers of Hepatitis B will be included in this study

               4. Voluntary written consent

        Exclusion Criteria:

          -  Pregnant or breastfeeding: Females of childbearing potential must have a blood test or
             urine study within 14 days prior to registration to rule out pregnancy

          -  Eligible for any higher priority transplant protocols

          -  Chemotherapy resistant disease

          -  Unrelated active infection