Donor Stem Cell Transplant With No or Low-Intensity Chemotherapy Using Sirolimus and Treated Immune Cells to Treat Blood and Lymph Cancers

* INCLUSION CRITERIA: PATIENT RECIPIENT 1. Patients with hematologic malignancies, myelodysplasia, or myeloproliferative disorders, as summarized in the following table. The diagnosis must be histologically confirmed by the Laboratory of Pathology of National Cancer Institute (NCI) or Hackensack (There will be no central pathology review). 2. Chronic Lymphocytic Leukemia - Disease Status: a) Relapse post-fludarabine, b) Non-Complete Response (CR) after salvage regimen. Hodgkin's and Non-Hodgkin's Lymphoma (all types, including Mantle Cell Lymphoma) - Disease Status: a) Primary treatment failure, b) Relapse after autologous stem cell transplant (SCT), c) Non-CR after salvage regimen Special Cases of High-Risk Lymphoma, including but not limited to : (1) plasma dendritic cell type, 2) Hepato-splenic T cell type, 3) gamma delta pinniculitic T cell type, 4) Muco-cutaneous natural killer (NK) cell type and 5) stage III-IV nasal NK cell type- Disease Status: a) Primary treatment failure, b) Relapse after autologous, c) Non-CR after salvage regimen, d) In first CR or any later CR Chronic Epstein Barr Virus (EBV)-associated lymphoproliferative disease a) At any point after diagnosis, including up-front therapy Multiple Myeloma - Disease Status: a) Primary treatment failure, b) Relapse after autologous stem cell transplant (SCT), c) Non-CR after salvage regimen. Acute Myelogenous Leukemia - Disease Status: a) CR number 1 and high-risk \[excludes t(8;21), t(15;17), or inv(16)\], b) CR number 2 or greater). Acute Lymphocytic Leukemia - Disease Status: a) CR number 1 plus high-risk \[t(9;22) or bcr-abl(+); t(4;11), 1(1;19), t(8;14)\], b) In CR number2 or greater. Myelodysplastic Syndrome - Disease Status: a) Refractory Anemia with Excess Blasts (RAEB), b) Refractory Anemia with Excess Blasts in Transformation (RAEB-T) (requires marrow and blood blasts less than 10% after induction chemotherapy). Myeloproliferative disorders - Disease Status: a) Idiopathic myelofibrosis, b) Polycythemia vera, c) Essential thrombocytosis, d) Chronic myelomonocytic leukemia. Chronic Myelogenous Leukemia (CML) - Disease Status: a) Chronic phase CML, refractory to imatinib treatment b) Accelerated phase CML. b) Accelerated phase CML Patients with myeloproliferative disorders must be end-stage, which is primarily defined as disease severity refractory to splenectomy. 3. Patient age of 16 to 75 years. 4. Consenting first degree relative matched at 6/6 HLA antigens (A, B, and DR). 5. Patient or legal guardian must be able to give informed consent. 6. All previous intravenous therapy administered outside of the National Institutes of Health (NIH) Clinical Center must be completed at least 2 weeks prior to study entry, with recovery to less than or equal to non-hematologic grade 2 toxicity of previous therapy. 7. Eastern Cooperative Oncology Group (ECOG) performance status equal to 0 or 1. 8. Life expectancy of at least 3 months. 9. Patients with acute leukemia must have chemotherapy sensitive disease, as defined by at least a 50% reduction in circulating absolute blast count due to the most proximal regimen. 10. Left ventricular ejection fraction greater than or equal to 45%, preferably by 2-dimension (2-D) echo, or by multi-gated acquisition scan (MUGA). However, patients with left ventricular ejection fraction (LVEF) of between 35% and 44% may also be eligible provided that such patients are cleared by a Cardiology Consultation that must include a cardiac stress test. 11. Corrected diffusing capacity or transfer of the lung for carbon monoxide (DLCO) greater than 50% of expected value. 12. Creatinine less than or equal to 1.5 mg/dl or creatinine clearance greater than or equal to 50 ml/min. 13. Serum total bilirubin less than 2.5 mg/dl; serum alanine aminotransferase (ALT) and aspartate aminotransferase (AST) equal 2.5 times upper limit of normal. Values above these levels may be accepted, at the discretion of the principal investigator (PI) or study chairman, if such elevations are thought to be due to liver involvement by malignancy or graft versus host disease (GVHD). 14. Adequate central venous access potential. 15. Potential patients referred for the study may not be eligible for the experimental protocol therapy due to reasons such as uncertainty about donor human leukocyte antigen (HLA) typing or need to control malignant disease, infection, or metabolic abnormality such as hypercalcemia on a emergent basis. Should a referred patient present to us in such a scenario, the patient will be referred back to their primary hematologist-oncologist for treatment. However, if referral back to the referring physician is not in the best interest of the patient according to the clinical judgement of the principal investigator (PI), then the patient may receive standard treatment for the malignant disease or complicating conditions (infection, metabolic problems under the current study. In other cases, a patient may have reasonable control of malignancy but does not meet the cluster of differentiation 4 (CD4) cell cut-off of 50 cells per microliter required for cohort 3 therapy; in such cases, standard care chemotherapy regimens may be administered for the specific goal of reducing the CD4 count (that is, immune depleting regimens such as the pentostatin plus cyclophosphamide combination, administered similar to the manner that we have developed on protocol 08-C- 0088). If it becomes apparent that the patient will not be able to proceed to experimental therapy, then he/she must come off study. Recipient-Subjects receiving a standard therapy, and availability of receiving the same treatment elsewhere, outside of a research protocol. Because such standard care therapy is not experimental, it is not necessary to complete the eligibility criteria prior to receiving such standard care; however, prior to initiation of the experimental therapy, the patient must meet each of the eligibility criteria detailed above. Attempts will be made to standardize such pretransplant chemotherapy (by administration of etoposide, prednisolone, oncovin, cyclophosphamide, hydroxydaunorubicin, fludarabine, rituximab (EPOCH-FR) chemotherapy, which is detailed later in this protocol); however, other regimens using approved agents will be allowed if such regimens are thought to be in the best interest of the patient. INCLUSION CRITERIA: DONOR 1. First-degree relative with genotypic identity at 6/6 HLA loci (HLA- A, B, and DR). 2. Age 11 to 90 years and able to give consent or assent. For donors \< 18 years old, the legal guardian must be able to provide informed consent. 3. Adequate venous access for peripheral apheresis, or consent to use a temporary central venous catheter for apheresis. 4. Donors must be human immunodeficiency virus (HIV) negative. 5. Donors with a history of hepatitis B or hepatitis C infection may be eligible. However, eligibility determination of such patients will require a hepatology consultation. The risk/benefit of the transplant and the possibility of transmitting hepatitis will be discussed with the patient and eligibility will then be determined by the principal investigator and lead associate investigator (LAI). 6. Lactating donors must substitute formula feeding for her infant during period of filgrastim administration (to prevent any filgrastim effect on infant). EXCLUSION CRITERIA: PATIENT 1. Active infection that is not responding to antimicrobial therapy. 2. Active central nervous system (CNS) involvement by malignancy. 3. HIV infection (treatment may result in progression of HIV and other viral infections). 4. Chronic active hepatitis B. Patient may be hepatitis B core antibody positive. For patients with concomitant positive hepatitis B surface antigen, patient will require a hepatology consultation. The risk/benefit profile of transplant and hepatitis B will be discussed with the patient and eligibility determined by the principal investigator and Lead Associate Investigator. 5. Hepatitis C infection. Patient may have hepatitis C infection. However, each patient will require a hepatology consultation. The risk/benefit profile of transplant and hepatitis C will be discussed with the patient and eligibility determined by the principal investigator and Lead Associate Investigator. 6. Pregnant or lactating. Patients of childbearing potential must use an effective method of contraception. The effects of the chemotherapy, the subsequent transplant and the medications used after the transplant are highly likely to be harmful to a fetus. The effects upon breast milk are also unknown and may be harmful to the infant. 7. History of psychiatric disorder which may compromise compliance with transplant protocol, or which does not allow for appropriate informed consent. EXCLUSION CRITERIA: DONOR 1. History of psychiatric disorder which may compromise compliance with transplant protocol, or which does not allow for appropriate informed consent. 2. History of hypertension that is not controlled by medication, stroke, autoimmune disease, or severe heart disease (donors with symptomatic angina will be excluded). Donors with a history of coronary artery bypass grafting or angioplasty who are symptom free will receive a cardiology evaluation and be considered on a case-by-case basis. 3. History of prior malignancy. However, cancer survivors who have undergone potentially curative therapy may be considered for stem cell donation on a case-by-case basis. In addition, donors with localized cancer such as prostate cancer that are on a watch-and-wait management due to the low-risk of disease progression may also be considered for stem cell donation on a case-by-case basis. The risk/benefit of the transplant and the possibility of transmitting viable tumor cells at the time of transplantation will be discussed with the patient. 4. Donors must not be pregnant (unknown effect of filgrastim on fetus). Donors of childbearing potential must use an effective method of contraception. 5. Anemia (Hemoglobin (Hb) less than 11 gm/dl) or thrombocytopenia (platelets less than 100,000 per microliter). However, potential donors with Hb levels less than 11 gm/dl that is due to iron deficiency will be eligible as long as the donor is initiated on iron replacement therapy and the case is individually approved by National Institutes of Health (NIH) or Hackensack Blood Bank.