Patients must not be candidates for allogeneic hematopoietic stem cell transplant; NOTE: Subjects up to age  years who are considered fit for allogeneic hematopoietic stem cell transplant, should be considered for enrollment on E, in order to avoid competing with that study; if a patient is considered unfit for intensive chemotherapy at the time of initial diagnosis, but subsequently achieves a complete remission (CR), then it will be left to the treating physicians discretion to consider hematopoietic stem cell transplant (HSCT)
Ineligible for hematopoietic stem cell transplant.
Hematopoietic stem cell transplant =<  months prior to registration
No prior hematopoietic transplant
Prior hematopoietic transplant
History of organ or hematopoietic stem cell transplant.
Patients who are not hematopoietic stem cell transplant candidates are excluded for the DLBCL cohort (cohort #)
Prior organ transplant including allogenic hematopoietic stem cell transplant
DONOR: Testing for communicable disease will be performed according to the University of Washington (UW) Hematopoietic Stem Cell Transplant Program guidelines set forth in the most current standard operating policies and procedures (UW Foundation for the Accreditation of Cellular Therapy [FACT]-accredited Clinical Hematopoietic Cell Processing Laboratory [CHCPL] standard of procedure [SOP]) for hematopoietic cell donor evaluation and selection
Prior hematopoietic stem cell transplant for AML
Be a recipient of hematopoietic stem cell transplant.
Patients who are hematopoietic stem cell transplant candidates are excluded
Prior hematopoietic stem cell transplant for the diagnosis of MDS
History of hematopoietic stem cell transplant (HSCT)
Has received allogeneic hematopoietic stem cell transplant within  months of CAR T cell infusion; hematopoietic stem cell transplant (HSCT) >  months from CAR T cell infusion eligible
Patient must be ?  months from hematopoietic stem cell transplant, must not have active GVHD, and must be off all immunosuppression
Received a hematopoietic stem cell transplant within the previous  months
Prior hematopoietic cell transplant: must be >=  months after previous transplant
Second hematopoietic cell transplant: Must be >=  months after prior myeloablative transplant
History of, or scheduled, hematopoietic stem cell transplant within  weeks of Screening
Subjects must be deemed ineligible for both high-dose chemotherapy and hematopoietic stem cell transplant (based on age, performance status and/or comorbidities) while also having adequate organ function for CAR T cell treatment.
Prior hematopoietic stem cell transplant
The subject must be a recipient of hematopoietic stem cell or solid organ transplant.
Hematopoietic cell transplant-co-morbidity Index greater than 
Has no other hematopoietic stem cell transplant of any type prior to the current planned autologous hematopoietic cell transplant
Has received any type of hematopoietic cell transplant
Subject is refractory to or relapsed after first-line AML therapy (with or without hematopoietic stem cell transplant (HSCT)).
Prior history of hematopoietic stem cell transplant
Prior hematopoietic stem cell transplant.
History of hematopoietic stem cell transplant
Prior allogeneic hematopoietic stem-cell transplant for participants with DLBCL, FL, MCL, and CLL only. Prior allogenic hematopoietic stem-cell transplant is permitted for participants with ALL
Newly diagnosed, symptomatic multiple myeloma patients for whom treatment is indicated per the NCCN guidelines, and for whom a hematopoietic stem cell transplant is not planned or scheduled during the study or are considered ineligible for hematopoietic stem cell transplant, with measurable disease
Received a hematopoietic stem cell transplant
Prior hematopoietic stem cell transplant
More than  months out from allogenic hematopoietic stem cell transplant prior to randomization.
Patient must not be a candidate for an allo-hematopoietic stem cell transplant (HSCT)
Transformed lymphoma is included if patients are ineligible for (or refuse) hematopoietic stem cell transplant
Patient has received any stem cell agents (other than hematopoietic graft) during study participation or within  days prior to study entry.
Has not received a prior hematopoietic stem cell transplant within the previous  months
Patients must be within    days after hematopoietic stem cell transplant (HSCT)
Subject has had hematopoietic stem cell transplant (HSCT) and meets any of the following:
Patients who are candidates for hematopoietic stem cell transplant
Prior treatment with hematopoietic stem cell transplant
Previous allogenic hematopoietic transplant within  d
Hematopoietic stem cell transplant recipient within  days post-transplant
Patients with previous hematopoietic stem cell transplant (HSCT)
Planned post-transplant maintenance therapy except for FLT inhibitors or TKIs must be declared prior to randomization. If it is known prior to enrollment that the hematopoietic stem cell product will need to be cryopreserved, the patient should not be enrolled.
Between  and  months after completion of cancer therapy (chemotherapy, radiation, hematopoietic cell transplant [HCT])
Has previously received an allogenic hematopoietic stem cell transplant.
Prior treatment with any adoptive T cell therapy; prior hematopoietic stem cell transplant (HSCT) is allowed
Has documented seropositivity for CMV within  year before hematopoietic stem cell transplant (HSCT)
Participant is candidate for hematopoietic stem cell transplants at the time of enrollment.