Obinutuzumab in cGVHD After Allogeneic Peripheral Blood Stem Cell Transplantation

Inclusion Criteria: * Subjects deemed potentially eligible by their treating physicians will be screened for enrollment after d+60 from transplantation * Patients who have undergone either ablative or non-myeloablative allogeneic stem cell transplantation are eligible. * Peripheral blood stem cells must have been used as the stem cell source. * Patients must have received transplantation from donors (both related and unrelated) who are identical at 8 HLA loci (A, B, C and DR1), or mismatched at no more than 1 locus (7/8). Among related donors, HLA C typing is not required (6/6 HLA matches). Class I typing is to be performed by PCR-SSP techniques and CDC techniques. Class II typing is performed by PCR-RFLP +/- PCR-SSP techniques. * No evidence of relapsed or residual malignancy within 30 days of trial entry. All patients must undergo appropriate staging for their malignancy (i.e. bone marrow aspiration for the Leukemias and PET-CT scanning for the lymphomas). Evidence of a persistent Cytogenetic abnormality will constitute evidence of residual or relapsed disease in the Leukemias, where present. Individuals with CLL are eligible if there is no more than 20% residual leukemia in the bone marrow at the time of study entry. * Patients who have undergone a non-myeloablative stem cell transplant must have \> 80% donor hematopoiesis within 30 days of study enrollment. Chimerism within 30 days of study entry must be greater than, equal to, or no more than 5% less than the chimerism measured at approximately day+30 (if performed). * Age ≥ 18.0 * ECOG performance status ≤2 (Karnofsky ≥60%) (See Appendix A) * Participants must have normal marrow function as defined by: * WBC ≥ 2,500/μL * Absolute Neutrophil Count ≥ 1,000/μL * Platelets ≥ 50,000/μL * Ability to understand and the willingness to sign a written informed consent document. * The effects of Obinutuzumab on the developing human fetus are unknown. For this reason, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 4 months after completion of Obinutuzumab administration. Exclusion Criteria: * Allogeneic stem cell transplantation using a single or multiple umbilical cord blood units or using bone marrow. * Allogeneic stem cell transplantation using in vivo or ex vivo T cell depletion, either by cell manipulation or with T cell depleting antibodies (Any anti-thymocyte globulin preparation or alemtuzumab given within 30 days of transplantation) * Participation in a clinical trial evaluating another preventative strategy for chronic GVHD, or ongoing participation in a clinical trial for therapy of acute GVHD. Prior completion of experimental therapy for acute GVHD is permissible if the experimental agent was used \> 30 days prior to enrollment. * Any evidence of ongoing gastrointestinal or hepatic acute GVHD, or evidence of greater than ongoing Stage I cutaneous acute GVHD. Ongoing, tapering therapy for resolved acute GVHD is permissible. * Any evidence of prior active or resolved chronic GVHD. * History of severe allergic reaction to Obinutuzumab * No Donor Lymphocyte Infusion (DLI) prior to day 100, and no plans for a DLI in the upcoming 30 days. * Evidence of any active uncontrolled infection (bacterial, viral or fungal) or evidence of natural exposure to Hepatitis B, Hepatitis C or HIV. Evidence of Hepatitis B exposure includes the presence of Hepatitis B surface antigenemia, a positive serological test for Hepatitis B core antibody or nucleic acid testing (NAT testing) that is positive for Hepatitis B. Vaccination to Hepatitis B is not an exclusion criteria. * Pregnancy or lactation. Negative pregnancy test is required within the screening window * Active use of any other investigational agents.