Reduced Intensity Conditioning Allogeneic Hematopoietic Stem Cell Transplant (HSCT) For Lymphoma

Inclusion Criteria: * Participants must have histologically or cytologically confirmed non-Hodgkin lymphoma involving the central nervous system (CNS) as defined by: * Biopsy of CNS mass in the brain, parenchyma, leptomeninges, or spinal cord demonstrating NHL. * Ocular biopsy from retina, subretinal pigment epithelial space, or optic nerve, or vitrectomy specimen, demonstrating NHL. * Biopsy of mass lesion outside of the CNS, or blood or body fluid specimen, documenting NHL, in conjunction with brain or spinal CT, PET/CT (positron emission tomography / computed tomography), or MRI showing radiographic abnormalities characteristic of CNS involvement with lymphoma. * CSF cytology demonstrating a malignant clonal NHL population, consistent with lymphomatous leptomeningitis, with or without a radiographically or pathologically identifiable CNS or systemic mass lesion. Patients with CSF (cerebral spinal fluid) studies negative for NHL by cytology but positive for a monoclonal population by flow cytometry and/or molecular PCR (polymerase chain reaction) studies may be eligible if they have radiographic evidence of a CNS lymphoma or if they have symptoms clinically consistent with CNS lymphomatous involvement; for such cases, please contact the protocol chair, Dr. Yi Bin Chen, to discuss eligibility prior to enrollment. * Patients must have experienced relapsed disease after high-dose chemotherapy with autologous stem cell transplantation (ASCT) OR have experienced relapse / progression on first-line high-dose methotrexate-based therapy and are not candidates for ASCT in the judgment of the treating physician. Discussion with the PI is encouraged for the latter scenario. * All participants must demonstrate a partial or complete response (PR or CR) of their CNS and systemic disease to pre-enrollment therapy and must be in PR or CR at the time of enrollment. Acceptable therapies include systemic or intrathecal chemo/immunotherapy and/or radiotherapy as well as corticosteroids. * Age ≥ 18 years as this study will be open at sites which treat adults. * ECOG (Eastern Cooperative Oncology Group) performance status ≤ 2 (Karnofsky score ≥ 60%, see Appendix A). * Participants must have adequate organ function as defined below: * Total serum bilirubin within normal institutional limits (unless patient has Gilbert's syndrome, where then direct serum bilirubin should be within normal institutional limits). * AST (SGOT) / ALT(SGPT) ≤ 3× institutional upper limit of normal. * Serum creatinine within normal institutional limits \--- OR * Creatinine clearance ≥ 50 mL/min/1.73 m2 for participants with creatinine level above institutional normal. * Left ventricular ejection fraction ≥ 40% measured either by echocardiogram or nuclear cardiac scan. * FEV1 (forced expiratory volume in 1 second), FVC (forced vital capacity) and DLCO all ≥ 50% of predicted (DLCO corrected for hemoglobin). * Participants must have a well-matched adult donor willing to donate peripheral blood stem cells with well-matched defined as: 8/8 matched related or unrelated donor (HLA-A, B, C, DRB1 by allele level). * Because chemotherapy agents as well as other therapeutic agents used in this trial are known to be teratogenic, 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 for 12 months after the date of stem cell transplantation. * Ability to understand and the willingness to sign a written informed consent document. * Eligibility Criteria for Donors * Donors must be medically fit to donate peripheral blood stem cells through standard G-CSF mobilization as assessed by institutional or unrelated marrow registry standards. * Donors will not have to sign a study specific informed consent to participate given that donors will be undergoing standard G-CSF mobilization and leukapheresis. * Exclusion Criteria: * Participants who have had chemotherapy or radiotherapy within 4 weeks prior to the 1st day of conditioning chemotherapy. * Participants who are receiving any other investigational agents. * History of allergic reactions attributed to compounds of similar chemical or biologic composition to fludarabine, thiotepa or other agents used in study. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements. * Pregnant or lactating women are excluded from this study because they are routinely ineligible to be treated with allogeneic stem cell transplantation. * HIV-positive participants on combination antiretroviral therapy are ineligible because of the potential for increased risk of lethal infections when treated with marrow-suppressive therapy.