Prior malignancy other than acute leukemia is allowed, provided it is in remission and there is no plan to treat the malignancy at the time of registration Prior malignancy Active malignancy (other than BRAF V mutation-negative melanoma) or malignancy within years A patient with a known other malignancy is eligible if there is a negligible risk for disease progression or death within one year, there is no active ongoing treatment for this malignancy, and the malignancy and/or any anticipated future treatments would not interfere with protocol-mandated evaluations at year Pts receiving active treatment or intervention for any other malignancy or pts who, at the Investigator's discretion, may require active treatment or intervention for any other malignancy within months of starting study treatment. Active malignancy other than the one for which AB- transplant is being performed within months of enrollment. Active cerebral/meningeal disease related to the underlying malignancy Other active malignancy that warrants systemic therapy Patients with suspected non-gynecologic malignancy, such as gastrointestinal Presence of an active hematological malignancy. No other active malignancy Persistence/relapse at the time of study entry of the primary malignancy for which the transplant was performed, or any other active malignancy; patients with a history of relapsed malignancy who have achieved a remission at the time of evaluation for study participation will not be excluded Patients must NOT have an active malignancy other than CD+ leukemia Previous malignancy is not an exclusion provided that the other malignancy is considered under control, patient is not on concomitant anti-cancer drug therapy, and target lesions from melanoma are clearly defined for response assessment Advanced measurable malignancy Previous malignancy is not an exclusion provided that the other malignancy is considered under control, patient is not on concomitant anti-cancer drug therapy, and target lesions from melanoma are clearly defined for response assessment No evidence of malignancy at the time of enrollment Any active malignancy other than DLBCL Other active malignancy T-cell malignancy Active malignancy (other than melanoma) or a prior malignancy within the past years No other active malignancy Subjects must NOT have an active malignancy other than CD+ leukemia No other active malignancy Evidence of active malignancy other than CD+ malignancy No other active malignancy Other active malignancy receiving systemic therapy. Active malignancy No active or co-existing malignancy requiring chemotherapy or radiation within months MALIGNANCY CRITERIA: Active CNS malignancy Have active malignancy with the exception of nonmelanoma skin cancer, as determined by the investigator. Subjects who experience relapse or progression of their underlying malignancy (for which HSCT was performed), as determined by the investigator, are not to be enrolled. Active cerebral/meningeal disease related to the underlying malignancy Presence of an active hematological malignancy: Presence of relapsed primary malignancy, or who have been treated for relapse after the alloHSCT was performed, or who may require rapid immune suppression withdrawal as pre-emergent treatment of early malignancy relapse. Does not have any other active malignancy other than the one for which this transplant is indicated Previous malignancy is not an exclusion provided that the other malignancy is considered under control, patient is not on concomitant anti-cancer drug therapy, and target lesions from melanoma are clearly defined for response assessment Active treatment of a separate malignancy Patients with prior malignancy successfully treated who are currently stable and on no active treatment are eligible Has a malignancy: No other active malignancy No active malignancy at the time of enrollment, except nonmelanoma skin cancers or carcinoma in situ; patients with a prior history of malignancy are eligible if their malignancy has been definitely treated or is in remission and does not require ongoing adjuvant or cancer-directed therapies Patient does NOT have an active malignancy other than NB Active and uncontrolled relapse of malignancy. Active and uncontrolled relapse of malignancy Active and uncontrolled relapse of malignancy Any other active malignancy Active and uncontrolled relapse of malignancy Uncontrolled, non-hematologic malignancy requiring active treatment RECIPIENT: Presence of active malignancy in another organ system other than the hematopoietic, except when driven by viruses in which case the immune reconstitution after transplant may control the malignancy Presence of malignancy. Patients receiving active treatment or intervention for any other malignancy or patients who, at the investigators discretion, may require active treatment or intervention for any other malignancy within months of starting study treatment Presence of active malignancy in another organ system other than the hematopoietic system, except when driven by viruses in which case the immune reconstitution after transplant may control the malignancy Resistant malignancy Subject has another malignancy for which treatment is required. Any malignancy Has another active systemic malignancy treated with chemotherapy within months before baseline (CD). Have active malignancy with the exception of nonmelanoma skin cancer. Subjects who have had a hematopoietic stem cell transplant (HSCT) and who experience relapse or progression of the malignancy as per investigator's opinion are not to be enrolled. Prior chemotherapy for any malignancy Other malignancy requiring active treatment Patient has another active malignancy Patients who have an active nd malignancy requiring systemic treatment Cerebral/meningeal disease related to the underlying malignancy that has not been definitively treated No other current active malignancy requiring anticancer therapy. Prior malignancy except for any of the following: Prior treatment for lymphoid malignancy: Active cerebral/meningeal disease related to the underlying malignancy. History of tuberous sclerosis, lymphangioleiomyomatosis (LAM) or any active malignancy Non-squamous malignancy of the penis Patients with another uncontrolled malignancy; patients with a previous malignancy, treated curatively and without evidence of disease relapse are eligible one or more of the following biomarkers of malignancy: Prior malignancy. Any active malignancy (other than BRAF-mutated melanoma) or a previous malignancy within the past years Cerebral/meningeal disease related to the underlying malignancy Other active malignancy Any other malignancy requiring treatment Any active malignancy within years that may alter the course of esophageal cancer; (apparently cured localized malignancy or advanced, but indolent malignancy with significantly more favorable prognosis are allowed) No active co-existing malignancy with life expectancy less than months due to that malignancy Advanced measurable malignancy another primary malignancy Another primary malignancy History of other malignancy (not including the underlying malignancy that was the indication for transplant) History of another active uncontrolled malignancy at the time of study enrollment No other active malignancy Active morphologic relapse/progression of hematologic malignancy post transplantation. Patients with more than one active malignancy at the time of enrollment. Cerebral/meningeal disease related to the underlying malignancy Subjects with another active malignancy The patient has a recently diagnosed active malignancy requiring therapy Subjects with no additional active malignancy within the last years Time since last therapy for treatment of underlying malignancy: No active or co-existing malignancy (other than ALL or lymphoblastic lymphoma) with life expectancy less than months due to that malignancy Presence of active malignancy from an organ system other than hematopoietic E . Prior malignancy. Patients must have indications for treatment for their B cell malignancy at the time of enrollment on this trial Other active malignancy Active malignancy with the exception of any of the following: Cerebral/meningeal disease related to underlying malignancy Active malignancy other than the one for which the UCB transplant is being performed within months of enrollment. Patients with more than one type of active malignancy; an active malignancy is defined as one that is being treated with therapeutic intent and for which survival may be impacted, within years of enrollment Other currently active malignancy History of other primary malignancy requiring systemic treatment within months of protocol enrollment; patients must not have another active malignancy requiring treatment; patients must not be receiving chemotherapy or immunotherapy for another cancer Other active malignancy Patients must not have another, non-breast, active malignancy that requires treatment Other active primary malignancy requiring treatment or limiting survival to =< years prior to registration Other active malignancy that requires therapy Cerebral/meningeal disease related to the underlying malignancy, unless definitively treated. Active and uncontrolled relapse of malignancy Confirmed malignancy. Does not have any other active malignancy other than the one for which this transplant is indicated Active and uncontrolled relapse of malignancy Patients receiving active treatment or intervention for any other malignancy or patients who, at the Investigator's discretion, may require active treatment or intervention for any other malignancy within months of starting study treatment. Malignancy OTHER than the BRAF mutant malignancy under study. Does not have any other active malignancy other than the one for which this transplant is indicated Patients who have a history of primary malignancy other than that being treated in this study, and currently requires active clinical intervention. Any other malignancy that requires active treatment History of any other malignancy requiring active treatment No other currently active malignancy. Other active malignancy. Other malignancy requiring active treatment at time of study entry Are undergoing cancer treatment for another malignancy Have a diagnosed malignancy Patients with evidence of persistent or active malignancy Another active malignancy Malignancy with bone instability Patients with active malignancy; patient undergoing active treatment for malignancy in the adjuvant setting are eligible; treatment can include chemotherapy, targeted therapy or anti-hormonal therapy Active and uncontrolled relapse of malignancy Active hematologic malignancy (cancer that begins in blood-forming tissue, such as leukemia or lymphoma) Active malignancy, metastatic disease, or anyone undergoing treatment for malignancy Women with an active malignancy Evidence of other active malignancy requiring treatment Patients may have had a prior malignancy