Active malignancy (other than BRAF V600 mutation-negative melanoma) or malignancy within 3 years Patients being actively treated for a secondary malignancy Active secondary malignancy Secondary malignancy. 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 1 year Active malignancy other than the one for which AB-110 transplant is being performed within 12 months of enrollment. Other active malignancy that warrants systemic therapy 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 CD19+ 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 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 Other active malignancy No active secondary malignancy No other active malignancy Subjects must NOT have an active malignancy other than CD22+ leukemia No other active malignancy Evidence of active malignancy other than CD19+ malignancy No other active malignancy Other active malignancy receiving systemic therapy. Active malignancy 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. 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 Patients with active secondary malignancy unless approved by the study chair 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 Patients with active secondary malignancy will not be eligible unless approved by the principal investigator Active and uncontrolled relapse of malignancy 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 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 Has another active systemic malignancy treated with chemotherapy within 12 months before baseline (C1D1). Patient has another active malignancy Any active secondary malignancy History of tuberous sclerosis, lymphangioleiomyomatosis (LAM) or any active malignancy Patients with another uncontrolled malignancy; patients with a previous malignancy, treated curatively and without evidence of disease relapse are eligible Other active malignancy Active secondary malignancy Any active malignancy within 3 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 12 months due to that malignancy History of another active uncontrolled malignancy at the time of study enrollment Patients must not have an active secondary malignancy No other active malignancy Patients with more than one active malignancy at the time of enrollment. Subjects with another active malignancy No active or co-existing malignancy (other than ALL or lymphoblastic lymphoma) with life expectancy less than 12 months due to that malignancy Presence of active malignancy from an organ system other than hematopoietic Other active malignancy Active malignancy with the exception of any of the following: Subjects being actively treated for a secondary malignancy. Active treatment for a secondary malignancy Secondary malignancy, with the exception of: Has active secondary malignancy that requires treatment. Undergoing active treatment for a secondary malignancy. Active malignancy other than the one for which the UCB transplant is being performed within 12 months of enrollment. Active malignancy besides HNSCC or primary skin basal cell carcinoma; (patients with a concomitant malignancy that has not progressed within 12 months of study entry are eligible) 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 3 years of enrollment Subjects being actively treated for a secondary malignancy. Other currently active malignancy Other active malignancy Patients must not have another, non-breast, active malignancy that requires treatment Other active malignancy that requires therapy Active and uncontrolled relapse of malignancy Presence of an active secondary malignancy. Patients with active secondary malignancy will not be eligible Does not have any other active malignancy other than the one for which this transplant is indicated No other known active secondary primary malignancy Active and uncontrolled relapse of malignancy Patients with active secondary malignancy will not be eligible unless approved by the principal investigator (PI) Does not have any other active malignancy other than the one for which this transplant is indicated Any other malignancy that requires active treatment No other currently active malignancy. Other active malignancy. Untreated active, or inappropriately/inadequately treated breast malignancy, without surgical treatment Secondary malignancy Patients with evidence of persistent or active malignancy Another active malignancy 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 malignancy, metastatic disease, or anyone undergoing treatment for malignancy Women with an active malignancy Patients diagnosed with secondary hepatic malignancy