Inclusion Criteria:

          -  :

               -  Patients aged 18-74 years at initial referral with a suitably matched related or
                  unrelated donor who have provided their informed consent to participate in the
                  clinical trial.

               -  If post-pubertal, females agree to take hormonal therapy to suppress menses
                  unless a specific contra-indication to estrogen exists

        Diagnosis:

          -  Patients with CD20+ aggressive B cell NHL (DLBCL, large cell transformation of
             indolent NHL/CLL, or mantle cell) OR CD20+ indolent NHL/CLL. Relapsed disease must be
             biopsy proven and CD20 positivity must be demonstrated within the 12 months prior to
             protocol enrollment.

        Eligible patients with DLBCL NHL will:

          -  have relapsed disease following initial therapy but failed to mobilize or had bone
             marrow involvement and therefore are not suitable for an autologous transplant OR

          -  have high-intermediate or high-risk second-line age-adjusted International Prognostic
             Index score and be in 2nd CR/PR following an autologous transplant OR

          -  have failed an autologous transplant and be in PR or better after salvage
             chemotherapy.

        Eligible patients with transformed indolent NHL/CLL will:

        • have CR/PR of the large cell component of their disease after either salvage chemotherapy
        or an autologous transplant.

        Eligible patients with mantle cell NHL will:

          -  be high-risk such as p53 positivity and be in 1st CR/PR after initial therapy OR

          -  have relapsed disease following initial therapy and be in 2nd or 3rd CR/PR after
             salvage chemotherapy.

        Eligible patients with indolent B cell NHL (such as, but not limited to, follicular, small
        cell or marginal zone NHL) or CLL will:

        • have 1st or subsequent progression or primary refractory disease (pre-allograft
        cytoreduction necessary but CR/PR not required).

        Pre-allograft Salvage Chemotherapy:

          -  This can include a single autologous transplant using high dose chemotherapy
             conditioning if appropriate OR ≥ 2 cycles of intensive combination chemotherapy (e.g.
             RICE) as appropriate according to diagnosis and prior therapy.

          -  CLL patients who have received CAMPATH do not have to receive pre-allograft salvage
             chemotherapy.

        Timing of PBSCT:

        • Admission for PBSCT must be within 120 days of autologous transplantation OR 80 days of
        the last cycle of chemotherapy.

        Organ Function and Performance Status Criteria:

          -  Karnofsky score ≥ 70 %

          -  calculated creatinine clearance ≥ 50 mL/min OR if creatinine ≥ 1.2, a history of renal
             dysfunction, age > 50 years, prior transplant, and/or a single kidney, the patient
             must have a measured creatinine clearance (using 24 hour urine collection) ≥ 50 mL/min

          -  bilirubin < 2.5, AST/ALT ≤ 3 x upper limit of normal (unless benign congenital
             hyperbilirubinemia)

          -  pulmonary function (spirometry and corrected DLCO) ≥ 50% normal

          -  left ventricular ejection fraction ≥ 40%

          -  albumin ≥ 2.5. Donor HLA-compatible related donors

          -  Patients who have an HLA-matched or one allele mismatched related donor are eligible
             for entry on this protocol. This will include a healthy related donor who is
             genotypically or phenotypically matched at least 9/10 of the A, B, C, DRB1, and DQB1
             loci, as tested by high resolution.

        HLA-compatible Unrelated donors • Patients who do not have a related HLA-matched donor but
        have an unrelated donor who is matched at

        ≥ 9/10 (allele mismatch only) of the A, B, C, DRB1, and DQB1 loci, as tested by high
        resolution.

        Exclusion Criteria:

          -  Diagnosis: known negativity for CD20 pre-allograft; mantle cell or DLBCL NHL with
             progressive disease at allograft work-up

               -  Prior Therapy: prior allogeneic transplant (prior autologous transplant is
                  acceptable)

               -  Cytoreduction and timing of NMA PBSCT: patients unable to complete planned
                  cytoreduction due to therapy complications, or who undergo cytoreduction but are
                  unable to proceed to allografting within the defined time period, are ineligible
                  for allograft on protocol

               -  Active and uncontrolled infection at time of transplantation including active
                  infection with Aspergillus or other mold, or HIV infection

               -  Patients positive for Hepatitis B or C at risk for viral reactivation.

               -  Inadequate performance status/organ function

               -  Pregnant or breast feeding

               -  Patient or guardian unable to give informed consent or unable to comply with the
                  treatment protocol including appropriate supportive care, follow-up and research
                  tests.