Inclusion Criteria:

1. Male or female recipients must have histopathologically confirmed diagnosis of hematological or lymphatic malignancy in one of the following categories:
2. Acute Leukemia: Recipients must have acute leukemia in second or greater remission in relapse, or primary refractory disease. Acute leukemia (in first remission with poor risk factors and molecular prognosis; acute myelogenous leukemia (AML) with -5, -7, t(6:9), +8, -11q23 and Acute lymphoblastic leukemia (ALL) with Phil+ t(9:22), t(4:11) and secondary remission inclusive).
3. Chronic myelogenous leukemia: Chronic Myeloid Leukemia (CML) in accelerated phase, blast crisis or second chronic phase.
4. Myelodysplastic syndrome (in high and intermediate risk categories) - marrow blast > 10% on differential.
5. Non-Hodgkin's lymphoma in relapse
6. Refractory chronic lymphoid leukemia (CLL) - refractory to fludarabine based regimen, unrelated donor and haploidentical only
7. The recipient must be <=60 years old at time of registration.
8. The recipient must have a related donor haploidentical for human leukocyte antigen (HLA), A, B, C, or DR loci. They may be partial matched on the other haplotype.
9. Recovery from prior therapy, chemotherapy, or radiotherapy, as defined by: Eastern Cooperative Oncology Group (ECOG) performance status equal or less than 2; have recovered from the toxicity of prior major chemotherapy at the start of the preparative regimen on this protocol
10. Adequate cardiac and pulmonary function (Left ventricular ejection fraction (LVEF) >45%, Carbon Monoxide Diffusing Capacity (DL CO)>50% corrected for hemoglobin)
11. Serum creatinine <1.5 mg/dL or creatinine clearance >50 ml/min for those above serum creatinine of 1.5; serum bilirubin <2.0 mg/dL; Aspartate transaminase (AST)/alanine aminotransferase (ALT) <2* Upper limits of normal (ULN) (unless secondary to disease)
12. Females of childbearing potential must have a negative serum or urine beta-HCG test within three weeks of registration. Patients will be informed of the risk of not receiving adequate contraception.
13. No prior cancer within five years with the exception of surgically cured non-melanoma skin cancer or in situ cancer of the cervix
14. The recipient and/or the recipient's legal guardian must have been informed of the investigational nature of this study and have signed a consent form which is in accordance with Federal guidelines and the guidelines of the participating institution.
15. Donor age must be 4-80 years and weight greater than 20 kg.
16. Medical history and physical examination confirm good health status as defined by institutional standards
17. Seronegative for HIV Ag, HIV 1+2 Ab, Human T Cell Leukemia Virus (HTLV) I/II Ab, HbsAg, HbcAb (IgM [combination screening test] and IgG), HCV, RPR for syphilis within 30 days of apheresis collection - If positive for Hepatitis B or C or syphilis, the recipient must be notified - the recipient may proceed if PI, recipient and donor agree and there is no substitute donor
18. HLA matching criteria
19. Female donors of child-bearing potential must have a negative serum or urine beta-HCG test within three weeks of mobilization
20. Capable of undergoing leukapheresis, have adequate venous access, and be willing to undergo insertion of a central catheter should leukapheresis via peripheral vein be inadequate
21. Agreeable to second donation of PBPC (or a bone marrow harvest) should the patient fail to demonstrate sustained engraftment following the transplant
22. The donor, or legal guardian greater than 18 years of age, must have been informed of the investigational nature of this study and have signed a consent form in accordance with Federal Guidelines and the guidelines of the participating institution. If the donor is less than 18 years of age, parent or legal guardian consent must be obtained.
23. The prospective donor will be screened for cytomegalovirus (CMV) seroreactivity and a seronegative donor will be utilized if available when the patient is seronegative.

Exclusion Criteria:

1. Participation in other clinical trials which involve investigational drugs or devices that might influence the endpoints of this study
2. Evidence of active hepatitis (B and/or C) or cirrhosis
3. Neither the recipient nor the donor may be HIV positive
4. Presence of any other active, uncontrolled bacterial, viral or fungal infection.
5. Uncontrolled central nervous system (CNS) involvement with tumor cells
6. Documented allergy to murine proteins or iron dextran
7. The recipient is a lactating female or, if of child-bearing potential, is unwilling to implement adequate birth control.
8. Severe end-organ dysfunctions, particularly neurologic deficits detectable by clinical examination or significant intellectual impairment in metabolic disorders
9. Evidence of active infection (including urinary tract infection, or upper respiratory tract infection) or hepatitis (on screening).
10. Medical or physical reason which makes the donor unlikely to tolerate or cooperate with growth factor therapy and leukapheresis.
11. Factors that place the donor at increased risk for complications from leukapheresis or G-CSF therapy such as pulmonary hypertension, coronary artery disease, peripheral vascular disease, cerebral vascular disease.
12. Lactating female or, if of child-bearing potential, is unwilling to implement adequate birth control.
13. Donors who are hepatitis positive, Human T-cell lymphotropic virus type I (HTLVI) positive need consent of Principal Investigator and determination that this is the best donor.