Inclusion Criteria:

          1. Age 18.0 years or older at the time of enrollment on Segment A

          2. Patients with acute leukemia or chronic myelogenous leukemia with no circulating
             blasts and with less than 5% blasts in the bone marrow

          3. Patients with myelodysplasia/chronic myelomonocytic leukemia with no circulating
             blasts and with less than 10% blasts in the bone marrow (higher blast percentage
             allowed in MDS due to lack of differences in outcomes with <5% vs. 5-10% blasts in
             this disease)

          4. Patients with relapsed chronic lymphocytic leukemia/small lymphocytic lymphoma with
             chemosensitive disease at time of transplantation

          5. Patients with lymphoma [follicular lymphoma, Hodgkin lymphoma, diffuse large B cell
             lymphoma, mantle cell lymphoma, peripheral T-cell lymphoma, angioimmunoblastic T-cell
             lymphoma and anaplastic large cell lymphoma] with chemosensitive disease at the time
             of transplantation

          6. Planned reduced intensity conditioning regimen (see eligible regimens in Table 2.4a)

          7. Patients must have a related or unrelated peripheral blood stem cell donor as follows:

               1. Sibling donor must be a 6/6 match for Human Leukocyte Antigen-A (HLA)-A and -B at
                  intermediate (or higher) resolution, and -DRB1 at high resolution using DNA-based
                  typing, and must be willing to donate peripheral blood stem cells and meet
                  institutional criteria for donation.

               2. Unrelated donor must be a 7/8 or 8/8 match at HLA-A, -B, -C and -DRB1 at high
                  resolution using DNA-based typing. Unrelated donor must be willing to donate
                  peripheral blood stem cells and meet National Marrow Donor Program (NMDP)
                  criteria for donation.

          8. Cardiac function: Left ventricular ejection fraction at least 45%

          9. Estimated creatinine clearance acceptable per institutional guidelines

         10. Pulmonary function: Diffusing capacity of lung for carbon monoxide (DLCO) corrected
             for hemoglobin at least 40% and forced expiratory volume at one second (FEV1)
             predicted at least 50%

         11. Liver function acceptable per institutional guidelines

         12. Karnofsky Performance Score at least 60%

         13. Female patients (unless postmenopausal for at least 1 year before the screening visit,
             or surgically sterilized), agree to practice two (2) effective methods of
             contraception at the same time, or agree to completely abstain from heterosexual
             intercourse, from the time of signing the informed consent through 12 months
             post-transplant (see Section 2.6.4 for definition of postmenopausal)

         14. Male patients (even if surgically sterilized), of partners of women of childbearing
             potential must agree to one of the following: practice effective barrier contraception
             (see Section 2.6.4 for list of barrier methods), or abstain from heterosexual
             intercourse from the time of signing the informed consent through 12 months
             post-transplant

         15. Plans for the use of post-transplant maintenance therapy must be disclosed upon
             enrollment and must be used irrespective of the outcome of the randomization. Please
             note that THIS DOES NOT INCLUDE INVESTIGATIONAL AGENTS and maintenance therapy with
             investigational treatment requires approval by the study chairs.

         16. Voluntary written consent obtained prior to the performance of any study-related
             procedure that is not a part of standard medical care, with the understanding that
             consent may be withdrawn by the patient at any time without prejudice to future
             medical care.

        Exclusion Criteria:

          1. Prior allogeneic transplant

          2. Active central nervous system (CNS) involvement by malignant cells

          3. Patients with secondary acute myeloid leukemia arising from myeloproliferative
             disease, including chronic myelomonocytic leukemia (CMML)

          4. Patients with uncontrolled bacterial, viral or fungal infections (currently taking
             medication and with progression or no clinical improvement) at time of enrollment.

          5. Presence of clinically significant fluid collection (ascites, pleural or pericardial
             effusion) that interferes with methotrexate clearance or makes methotrexate use
             contraindicated

          6. Patients seropositive for human immunodeficiency virus (HIV) with detectable viral
             load. HIV+ patients with an undetectable viral load on antiviral therapy are eligible.

          7. Myocardial infarction within 6 months prior to enrollment or New York Heart
             Association (NYHA) Class III or IV heart failure, uncontrolled angina, severe
             uncontrolled ventricular arrhythmias, or electrocardiographic evidence of acute
             ischemia.

          8. Female patients who are pregnant (as per institutional practice) or lactating

          9. Patients with a serious medical or psychiatric illness likely to interfere with
             participation in this clinical study

         10. Patients with prior malignancies except resected non-melanoma skin cancer or treated
             cervical carcinoma in situ. Cancer treated with curative intent ≥ 5 years previously
             will be allowed. Cancer treated with curative intent < 5 years previously must be
             reviewed and approved by the Protocol Officer or Chairs.

         11. Planned use of antithymocyte globulin (ATG) or alemtuzumab in conditioning regimen