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