Autologous or Donor Stem Cell Transplantation in Treating Patients With Recurrent Non-Hodgkin's Lymphoma (BMT CTN 0202)

Initial Patient Inclusion Criteria: * Histologically confirmed recurrent Revised European American Lymphoma (REAL) classification follicle center lymphoma, follicular grades I and II, OR histologically confirmed World Health Organization (WHO) classification follicular lymphoma grades 1, 2, 3a or 3b; for either classification, the diffuse component or presence of large cleaved cells (if present) cannot be more than 50% of high power field; patients do not have to express t(14;18) to be eligible * Received three or fewer prior regimens of chemotherapy; monoclonal antibody therapy and involved field radiation therapy will not be counted as a prior therapy * Beyond first Complete Remission (CR) or first Partial Remission (PR) AND demonstrate chemosensitive disease; chemosensitive disease will be defined as less than 20% bone marrow involvement in the aspirate or core biopsy with follicular lymphoma AND lymph node size in axial diameter of less than 3 cm or a greater than 50% reduction in estimated lymph node volume to be measured as product of bi-dimensional measurements; Positron Emission Tomography (PET) scanning will not be used for staging or response purposes * Patients with adequate organ function as measured by: 1. Cardiac: left ventricular ejection fraction at rest at least 45% 2. Hepatic: bilirubin less than 2 times the upper limit of normal and alanine transaminase (ALT) and aspartate aminotransferase (AST) less than 3 times the upper limit of normal 3. Renal: creatinine clearance greater than 40 mL/min 4. Pulmonary: Diffusing capacity of the lung for carbon monoxide (DLCO), Forced expiratory volume in one second (FEV1), and Forced vital capacity (FVC) greater than 50% of predicted (corrected for hemoglobin) * If the patient is younger than 18 years of age and they have reached the age of assent, then they must have completed the local Institutional Review Board (IRB) assent process. * Able to receive cyclophosphamide and rituximab mobilization chemotherapy no earlier than 3 weeks from the beginning of the most recent cycle of salvage chemotherapy and no later than 6 weeks from enrollment Patient Inclusion Criteria for Proceeding to Hematopoietic Stem Cell Transplant (HSCT): * Collection of an autologous or allogeneic graft of at least 2.0 \* 10\^6 CD34+ cells/kg * Blood count recovery defined as Absolute Neutrophil Count (ANC) greater than 1000/mm3 and platelets greater than 100 \* 10\^9/L Patient Inclusion Criteria for Maintenance Therapy: * Liver and renal function tests within the inclusion criteria for initial autograft * Off intravenous antibiotics and off amphotericin B formulations for proven, probable or possible fungal infections * No active Cytomegalovirus (CMV) infections or for patients with CMV infection post-autograft, treated with ganciclovir, valganciclovir, or foscarnet per institutional guidelines and CMV antigenemia negative * Mucositis resolved and off hyperalimentation Exclusion Criteria: * Karnofsky performance score less than 70% * Follicular lymphoma that show histologic evidence of transformation * Uncontrolled hypertension * Patients with uncontrolled bacterial, viral or fungal infection (currently taking medication and progression without clinical improvement). * Prior malignancies except resected basal cell carcinoma or treated cervical carcinoma in situ; cancer treated with curative intent more than 5 years previously will be reviewed on a case-by-case basis by a Protocol Chair or Medical Monitor. * Pregnant (positive Beta Human chorionic gonadotropin (β-HCG)) or breastfeeding * Seropositive for Human immunodeficiency virus (HIV) * Unwilling to use contraceptive techniques during treatment * Prior autologous or allogeneic HSCT * Known anaphylactic reaction to rituximab