Inclusion Criteria:

          -  Must have a human leukocyte antigen (HLA)-matched or single allele-mismatched adult
             sibling serving as donor

          -  Must have a myeloid or lymphoid malignant disease that is treated with TLI and ATG
             reduced intensity conditioning for allogeneic transplant (any of the following AML,
             myelodysplastic syndrome [MDS], myeloproliferative disease [MPD], CLL, B or T-cell
             NHL, HL)

          -  Patients who due to age, pre-existing medical conditions, or, prior therapy are
             considered to be at high risk for regimen related toxicity associated with fully
             ablative transplant conditioning, and therefore reduced intensity conditioning is
             recommended

          -  Ability to understand and the willingness to sign a written informed consent document;
             patients must have signed informed consent to participate in the trial

          -  DONOR: Must be an HLA-matched or single allele mismatched sibling of enrolled
             transplant patient

          -  DONOR: Must be 18-75 years of age, inclusive

          -  DONOR: Must be in a state of general good health and have completed a donor evaluation
             with history, medical examination and standard blood tests within 35 days of starting
             the hematopoietic cell collection procedure; in order to fairly represent the
             interests of the donor, the donor evaluation and consent will be performed by a study
             team member other than the recipient's attending physician

          -  DONOR: Must have a white blood cell count > 3.5 x 10^9/liter, platelets > 150 x
             10^9/liter and hematocrit > 35%

          -  DONOR: Must be capable of undergoing leukapheresis

          -  DONOR: Must be able to understand and sign informed consent

          -  DONOR: Must not be seropositive for HIV 1 and 2, hepatitis B surface antigen,
             hepatitis C antibody, human T-lymphotropic virus (HTLV) antibody, cytomegalovirus
             (CMV) immunoglobulin M (IgM), or rapid plasma reagin (RPR) (Treponema); donors with
             prior evidence of hepatitis B core antibody positivity will have a polymerase chain
             reaction (PCR) test done to evaluate for hepatitis B infection; donors with a positive
             hepatitis B PCR test are excluded

          -  DONOR: Females must not be pregnant or lactating

          -  DONOR: Must not have psychological traits or psychological or medical conditions which
             make them unlikely to tolerate the procedure

          -  DONOR: Must not have developed a new malignancy requiring chemotherapy or radiation in
             the interval since apheresis for initial hematocrit (HCT)

          -  PATIENT CRITERIA FOR PROCEEDING WITH CD8+ MEMORY T-CELL INFUSION:

          -  Patients must be beyond day 30 and before day 60 after transplant

          -  Patients must have evidence of mixed CD3 T-cell chimerism based on the day +28 (+/- 7
             days) blood sample showing >= 5% and =< 95% donor type cells

          -  Patients must have no evidence of active graft-versus-host disease at the time of the
             CD8+ memory T-cell infusion; patients with a history of acute GVHD overall grade II
             based on skin only involvement or upper gastrointestinal (GI) tract involvement only
             will be eligible; patients with a history of liver or lower GI tract GVHD will not be
             eligible

          -  Patients must be on single immune suppression therapy with either tacrolimus or
             cyclosporine at the time of CD8+ memory T-cell infusion; prednisone at a physiologic
             dose of 5 mg per day or less is allowed

          -  Patients must have a Karnofsky performance status of >= 60% at the time of the CD8+
             memory T-cell infusion

          -  Patients must not have an uncontrolled bacterial, fungal or viral infection, defined
             as progressive symptoms despite therapy, at the time of the CD8+ memory T-cell
             infusion; asymptomatic viremia is allowed

          -  Patients must have adequate organ function and performance status at the time of the
             CD8+ memory T-cell infusion, defined by the following:

               -  Total bilirubin =< 4 mg/dL

               -  SGOT or SGPT =< 4 x ULN

               -  Creatinine =< 3 mg/dL or estimated creatinine clearance >= 40ml/min

        Exclusion Criteria:

          -  Uncontrolled bacterial, viral or fungal infection defined as currently taking
             medication and progression of clinical symptoms

          -  Progressive hemato-lymphoid malignancy despite conventional therapy

          -  Acute leukemia not in remission

          -  Chronic myelogenous leukemia (CML)

          -  Active central nervous system (CNS) involvement of the underlying malignancy

          -  Human immunodeficiency virus (HIV) positive

          -  Pregnant or lactating

          -  Prior malignancy (EXCEPTION: diagnosed > 5 years ago without evidence of disease, OR
             treated =< 5 years ago but have a greater than 50% chance of life expectancy of >= 5
             years for that malignancy)

          -  Have a psychiatric disorder(s) or psychosocial circumstance(s) which in the opinion of
             the primary physician would place the patient at an unacceptable risk from transplant

          -  Ejection fraction < 30%, or uncontrolled cardiac failure

          -  Diffusing capacity of the lung for carbon monoxide (DLCO) < 40% predicted

          -  Total bilirubin > 3 mg/dL

          -  Serum glutamic oxaloacetic transaminase (SGOT) or serum glutamate pyruvate
             transaminase (SGPT) > 4 x upper limit of normal (ULN)

          -  Creatinine > 2 mg/dL and an estimated creatinine clearance =< 40 mL/min

          -  Poorly controlled hypertension despite multiple antihypertensive medication OR

          -  Karnofsky performance status (KPS) < 60%

          -  Note: Patients positive for hepatitis B and C will be evaluated on a case by case
             basis