Inclusion Criteria:

- Ages > 50 years with hematologic malignancies treatable by unrelated hematopoietic
cell transplant (HCT)

- Ages =< 50 years of age with hematologic diseases treatable by allogeneic HCT who
through pre-existing medical conditions or prior therapy are considered to be at high
risk for regimen related toxicity associated with a conventional transplant (> 40%
risk of transplant related mortality [TRM]) (This criterion can include patients with
a HCT-comorbidity index (CI) score of >= 1; transplants should be approved for these
inclusion criteria by both the participating institutions' patient review committees
such as the Patient Care Conference (PCC) at the Fred Hutchinson Cancer Research
Center (FHCRC) and by the principal investigators at the collaborating centers)

- Patients =< 50 years of age who have received previous high-dose transplantation do
not require patient review committee approvals (All children < 12 years must be
discussed with the FHCRC principal investigator (PI) [Brenda Sandmaier, MD 206
6674961] prior to registration)

- Ages =< 50 years of age with chronic lymphocytic leukemia (CLL); these patients do
not require patient review committee approvals

- Ages =< 50 years of age with hematologic diseases treatable by allogeneic HCT who
refuse a conventional HCT (Transplants must be approved for these inclusion criteria
by both the participating institutions' patient review committee such as PCC at the
FHCRC and by the principal investigators at the collaborating centers)

- The following diseases will be permitted although other diagnoses can be considered
if approved by PCC or the participating institutions' patient review committees and
the principal investigators:

- Aggressive non-Hodgkin lymphomas (NHL) and other histologies such as Diffuse
large B cell NHL not eligible for autologous hematopoietic stem cell transplant
(HSCT), not eligible for conventional myeloablative HSCT, or after failed
autologous HSCT

- Mantle Cell NHL may be treated in first complete response (CR) (Diagnostic
lumbar puncture [LP] required pretransplant)

- Low grade NHL with < 6 month duration of CR between courses of conventional
therapy

- Multiple Myeloma must have received prior chemotherapy (Consolidation of
chemotherapy by autografting prior to nonmyeloablative HCT is permitted)

- Acute Myeloid Leukemia (AML ) must have < 5% marrow blasts at the time of
transplant

- Acute Lymphocytic Leukemia (ALL) must have < 5% marrow blasts at the time of
transplant

- Chronic Myeloid Leukemia (CML) patients will be accepted if they are beyond
Chronic Phase (CP)1 and if they have received previous myelosuppressive
chemotherapy or HCT and have < 5% marrow blasts at time of transplant

- Myelodysplasia (MDS)/Myeloproliferative Syndrome (MPS) patients must have
received previous myelosuppressive chemotherapy or HCT and have < 5% marrow
blasts at time of transplant

- Waldenstrom's Macroglobulinemia must have failed 2 courses of therapy

- CLL must have either

- 1) failed to meet National Cancer Institute (NCI) Working Group criteria for
complete or partial response after therapy with a regimen containing FLU
(fludarabine phosphate) (or another nucleoside analog, e.g. Cladribine [2-CDA],
pentostatin) or experience disease relapse within 12 months after completing
therapy with a regimen containing FLU (or another nucleoside analog)

- 2) failed FLU-CY (cyclosporine)-Rituximab (FCR) combination chemotherapy at any
time point; or

- 3) have 17p deletion cytogenetic abnormality (Patients should have received
induction chemotherapy but could be transplanted in 1st CR)

- The following diseases are permitted:

- Hodgkin Lymphoma must have received and failed frontline therapy

- Multiple Myeloma must have received prior chemotherapy (Consolidation of
chemotherapy by autografting prior to nonmyeloablative HCT is permitted)

- Acute Myeloid Leukemia (AML ) must have < 5% marrow blasts at the time of
transplant

- Acute Lymphocytic Leukemia (ALL) must have < 5% marrow blasts at the time of
transplant

- Chronic Myeloid Leukemia (CML) patients will be accepted if they are beyond
Chronic Phase (CP)1 and if they have received previous myelosuppressive
chemotherapy or HCT and have < 5% marrow blasts at time of transplant

- Myelodysplasia (MDS)/Myeloproliferative Syndrome (MPS) patients must have
received previous myelosuppressive chemotherapy or HCT and have < 5% marrow
blasts at time of transplant

- Waldenstrom's Macroglobulinemia must have failed 2 courses of therapy

- Diffusion capacity of carbon monoxide (DLCO) < 40%, total lung capacity (TLC) < 40%,
forced expiratory volume in one second (FEV1) < 40% and/or receiving supplementary
continuous oxygen

- DONOR: FHCRC matching allowed will be Grades 1.0 to 2.1: unrelated donors who are
prospectively:

- Matched for human leukocyte antigen (HLA)-A, B, C, DRB1 and DQB1 by high
resolution typing

- Only a single allele disparity will be allowed for HLA-A, B, or C as defined by
high resolution typing

- DONOR: Donors are excluded when preexisting immunoreactivity is identified that would
jeopardize donor hematopoietic cell engraftment; this determination is based on the
standard practice of the individual institution; the recommended procedure for
patients with 10 of 10 HLA allele level (phenotypic) match is to obtain a panel
reactive antibody (PRA) screens to class I and class II antigens for all patients
before HCT; if the PRA shows > 10% activity, then flow cytometric or B and T cell
cytotoxic cross matches should be obtained; the donor should be excluded if any of
the cytotoxic cross match assays are positive; for those patients with an HLA Class I
allele mismatch, flow cytometric or B and T cell cytotoxic cross matches should be
obtained regardless of the PRA results; a positive anti-donor cytotoxic crossmatch is
an absolute donor exclusion

- DONOR: Patient and donor pairs homozygous at a mismatched allele in the graft
rejection vector are considered a two-allele mismatch, i.e., the patient is A*0101
and the donor is A*0102, and this type of mismatch is not allowed

- DONOR: Only filgrastim (G-CSF) mobilized peripheral blood mononuclear cell (PBMC)
only will be permitted as a HSC source on this protocol

Exclusion Criteria:

- Patients with rapidly progressive intermediate or high grade NHL

- Patients with a diagnosis of chronic myelomonocytic leukemia (CMML)

- Central nervous system (CNS) involvement with disease refractory to intrathecal
chemotherapy

- Presence of circulating leukemic blasts (in the peripheral blood) detected by
standard pathology for patients with AML, MDS, ALL or CML

- Fertile men or women unwilling to use contraceptive techniques during and for 12
months following treatment

- Females who are pregnant or breast-feeding

- Patients with active non-hematological malignancies (except non-melanoma skin
cancers) or those with non-hematological malignancies (except non-melanoma skin
cancers) who have been rendered with no evidence of disease, but have a greater than
20% chance of having disease recurrence within 5 years

- Fungal infections with radiological progression after receipt of amphotericin B or
active triazole for greater than 1 month

- Cardiac ejection fraction < 35% (Ejection fraction is required if age > 50 years or
there is a history of anthracycline exposure or history of cardiac disease)

- The FHCRC PI of the study must approve of enrollment of all patients with pulmonary
nodules

- Patients with clinical or laboratory evidence of liver disease would be evaluated for
the cause of liver disease, its clinical severity in terms of liver function, and the
degree of portal hypertension; patients will be excluded if they are found to have
fulminant liver failure, cirrhosis of the liver with evidence of portal hypertension,
alcoholic hepatitis, esophageal varices, a history of bleeding esophageal varices,
hepatic encephalopathy, uncorrectable hepatic synthetic dysfunction evinced by
prolongation of the prothrombin time, ascites related to portal hypertension,
bridging fibrosis, bacterial or fungal liver abscess, biliary obstruction, chronic
viral hepatitis with total serum bilirubin > 3 mg/dL, or symptomatic biliary disease

- Karnofsky score < 60 or Lansky score < 50

- Patient has poorly controlled hypertension and on multiple antihypertensives

- Human immunodeficiency virus (HIV) positive patients

- Active bacterial or fungal infections unresponsive to medical therapy

- All patients receiving antifungal therapy voriconazole, posaconazole, or fluconazole
and who are then randomized to ARM 3 must have rapamycin reduced according to the
Standard Practice of Antifungal Therapy Guidelines

- The addition of cytotoxic agents for cytoreduction with the exception of tyrosine
kinase inhibitors (such as imatinib), cytokine therapy, hydroxyurea, low dose
cytarabine, chlorambucil, or rituxan will not be allowed within three weeks of the
initiation of conditioning

- DONOR: Donor (or centers) who will exclusively donate marrow

- DONOR: Donors who are HIV-positive and/or, medical conditions that would result in
increased risk for G-CSF mobilization and harvest of G-PBMC