Patient must not have an antecedent hematologic disorder Ages > 50 years with hematologic malignancies treatable by related or unrelated HCT Acceptable hematologic status: Acceptable hematologic status: Platelets >= 75,000/mcL (>= 50,000 for patients with hematologic malignancies) Histologically documented metastatic cancer (solid tumors, not including hematologic malignancies) Acceptable hematologic status Participants with hematologic lymphoma Histologically documented metastatic cancer (solid tumors, not including hematologic malignancies) Participants with hematologic malignancies Acceptable hematologic status at Screening Have current hematologic malignancies, acute or chronic leukemia. Evidence of relapse or progression of hematologic malignancy at the time of study enrollment. Confirmed advanced hematologic malignancies; Phase 1: Hematologic malignancies other than lymphomas. Cold agglutinin syndrome secondary to infection, rheumatologic disease, or active hematologic malignancy Hematologic malignancies other than lymphomas. Ages > 50 years with hematologic malignancies treatable by unrelated hematopoietic cell transplant (HCT) Any history of or current hematologic malignancy Current hematologic malignancies. Concurrent hematologic or non-hematologic malignancy requiring treatment Patients with life expectancy of =< 6 months for reasons other than their underlying hematologic/oncologic disorder Patients must be diagnosed with a high-risk and/or advanced hematologic malignancy defined as one of the following Germ cell or hematologic malignancies Hematologic malignancy Loss of complete hematologic response (CHR) Patients with history of hematologic malignancy Patients with life expectancy of =< 6 months for reasons other than their underlying hematologic/oncologic disorder Concurrent hematologic or non-hematologic malignancy requiring treatment (other than multiple myeloma and secondary amyloidosis) Any history of or current hematologic malignancy INCLUSION CRITERIA FOR ENROLLMENT: Active relapse involving the bone marrow of a hematologic malignancy >= 6 months after allogeneic hematopoietic cell transplant (alloHCT) employing PTCy as GVHD prophylaxis Remission of any acute hematologic malignancy or adequate disease control for chronic malignancies Treatment for non-hematologic malignancy greater than 6 months prior to enrollment is acceptable Patients must have histo-pathologically confirmed diagnosis of hematologic malignancy (leukemia or lymphoma); patients to be considered will either have resistant/refractory hematologic malignancies (disease exceeding 5% by morphology or with measurable extramedullary diseases: e.g. nodal disease or chloroma) or have relapsed following an initial allogeneic HSCT; alternatively, they will have a hematologic malignancy where allogeneic HSCT is indicated and their disease is in a state of remission, but they lack a well matched donor: related, unrelated donor (URD) or umbilical cord blood (UCB) HSC source (less than 7 of 8 URD match, less than 4 of 6 UCB match and/or less than 2 x10^5 CD 34+ HSC/kg in a matched UCB product); patients between 19 and 24 years of age will have been previously diagnosed and cared for by a Pediatric Oncologist Uncontrolled, non-hematologic malignancy requiring active treatment Hematologic or lymphoid malignancy Patients with life expectancy of =< 6 months for reasons other than their underlying hematologic/oncologic disorder Acceptable hematologic status Patients with life expectancy of =< 6 months for reasons other than their underlying hematologic/oncologic disorder Patients with life expectancy of =< 6 months for reasons other than their underlying hematologic/oncologic disorder Hematologic malignancies other than lymphomas. Cold agglutinin syndrome secondary to infection, rheumatologic disease, or active hematologic malignancy Other hematologic/biochemistry requirements, as per protocol Patient must have hematologic malignancy that meets institutional eligibility requirements for cord blood transplant Patients receiving any other standard or investigational treatment for their hematologic malignancy Concurrent hematologic or non-hematologic malignancy requiring treatment (other than multiple myeloma or secondary amyloidosis) Untreated first hematologic relapse is defined as: Subject is in second or later hematologic relapse or has received salvage therapy for refractory disease Part 1: solid tumors or lymphomas, or hematologic malignancies Acceptable hematologic status: Diagnosis of any of the following hematologic malignancies: Hematologic and Non-hematologic exclusion criteria before start of therapy. Patients in CRi must have evidence of hematologic recovery after prior therapy to at least: platelets >= 75 x 10^9/L Patients with history of hematologic malignancies who have received at least 2 lines of standard chemoimmunotherapy and have persistent disease ?1 prior systemic hematologic therapy for a free light chain (FLC) producing hematologic malignancy underlying the initial diagnosis of AL amyloidosis with at least a partial FLC response (PR, VGPR, CR) to treatment deemed stable and not requiring further treatment Patients with high risk hematologic malignancies, including those with induction failure and in relapse Presence of < 4% blasts on hematologic studies Patients with life expectancy of =< 6 months for reasons other than their underlying hematologic/oncologic disorder Active morphologic relapse/progression of hematologic malignancy post transplantation. Splenomegaly or history of proliferative hematologic disease Ages > 50 years with hematologic malignancies treatable by unrelated hematopoietic cell transplant (HCT) Patients with the following hematologic malignancies will be permitted although other diagnoses can be considered if approved by Patient Care Conference (PCC) and the principal investigators: Diagnosis of hematologic malignancy and at least one of the criteria as specified above Subjects must have a histopathologically confirmed diagnosis of an advanced non-hematologic malignancy or lymphoma or indolent NHL/CLL. Patients with life expectancy of =< 6 months for reasons other than their underlying hematologic/oncologic disorder or complications there from Concurrent non-hematologic malignancies requiring treatment. Histopathologically confirmed diagnosis of a metastatic non-hematologic malignancy. For the hematologic malignancy patients, blood count values cited above do not apply. Other high risk hematologic malignancies to be approved by 2 or more hematology/oncology and bone marrow transplant (BMT) physicians ?1 prior systemic plasma cell dyscrasia therapy with at least a partial hematologic response Any prior history of other hematologic malignancy besides FL or myelodysplasia Newly Diagnosed Secondary AML age <60 years and ?76 to 80 years, defined as having a history of an antecedent hematologic disorder (myelodysplastic syndromes [MDS], myeloproliferative disease [MPD]or history of cytotoxic treatment for non-hematologic malignancy) Evidence of significant hepatic, hematologic, or immunologic disease. Antecedent hematologic disease Hematologic: Receiving any other standard treatment for their hematologic malignancy. Hematologic status: Therapy with any other standard or investigational treatment for hematologic malignancy (except hydroxyurea, as mentioned in the inclusion criteria) Uncontrolled concurrent illness including: other hematologic or non-hematologic malignancy, active infection, or uncontrolled diabetes Have known current hematologic malignancies or acute or chronic leukemia Newly Diagnosed Secondary AML defined as having a history of an antecedent hematologic disorder (myelodysplastic syndromes [MDS], myeloproliferative disease [MPD]or history of cytotoxic treatment for non-hematologic malignancy) Patients with history of hematologic disease, including myelodysplasia or bone marrow malignancies A diagnosis of any hematologic or solid malignancy Phase I: Having a non-hematologic malignancy reason for undergoing transplantation (e.g. aplastic anemia) Admission to the MSKCC Adult BMT service for an outpatient HSCT for a hematologic malignancy Patients with hematologic malignancy receiving care (chemotherapy, immunotherapy, targeted agents, bone marrow transplant, or other) for their hematologic malignancy at the Ohio State University Hematologic malignancy Hematologic malignancies including acute and chronic leukemias, myelodysplastic syndromes, and myeloma Patients diagnosed with a hematologic malignancy Adult patients with hematologic malignancy admitted to Massachusetts General Hospital (MGH) HSCT are eligible for the study A total bilirubin of less than 2.5 mg/dL, unless these values are due to underlying hematologic malignancy Received a transplant at a consortium center for a hematologic malignancy or myelodysplasia Active hematologic malignancy (cancer that begins in blood-forming tissue, such as leukemia or lymphoma) Evidence of hematologic malignancy or disease relapse post-transplant (stable mixed chimerisms is permitted) FOR THE 31 SUBJECTS ENROLLED IN YEAR 1: Evidence of hematologic malignancy or disease relapse post-transplant (stable mixed chimerisms is permitted) Evidence of hematologic malignancy or disease relapse post-transplant (stable mixed chimerisms are permitted) EXCLUSION CRITERIA FOR ENROLLMENT (PRE-TRANSPLANT): Non-hematologic malignancy with a life expectancy of < 5 years Patients must have a hematologic malignancy treatable by nonmyeloablative HCT; the following diseases will be permitted although other diagnoses can be considered if approved by Patient Care Conference (PCC) and the principal investigator: Any of the following hematologic malignancies: Hematologic malignancies with the exception of lymphoma Any of the following hematologic malignancies: Non-hematologic malignancy within prior three (3) years. No current or history of a hematologic malignancy is allowed, including subjects who have undergone a bone marrow transplant. 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