Safety and Efficacy of ALLO-501A Anti-CD19 Allogeneic CAR T Cells in Adults With Relapsed/Refractory Large B Cell Lymphoma, Chronic Lymphocytic Leukemia and Small Lymphocytic Lymphoma (ALPHA2)

Inclusion Criteria:

For subjects with LBCL:

* Histologically confirmed diagnosis of relapsed/refractory large B-cell lymphoma at last relapse per WHO 2017
* At least 1 measurable lesion at time of enrollment
* Relapsed or refractory disease after at least 2 lines of chemotherapy
* Absence of significant donor (product)-specific anti-HLA antibodies (DSA) at screening (Note: Only applicable for Phase 2)

For subjects with CLL/SLL:

* Diagnosis of CLL/SLL
* Relapsed/refractory disease
* Subjects relapsed/refractory to BTKi therapy and high-risk disease
* Subjects relapsed/refractory with 2 or more lines of therapy including BTKi and BCL-2 inhibitor (venetoclax)
* At least 1 measurable lesion at time of enrollment

For all subjects:

* Male or female subjects ≥18 years of age
* Eastern Cooperative Oncology Group (ECOG) performance status (PS) of 0 or 1
* Adequate hematological, renal, and liver function

Exclusion Criteria:

* Active central nervous system (CNS) involvement by malignancy
* Current thyroid disorder (including hyperthyroidism), except for subjects with hypothyroidism controlled on a stable dose of hormone replacement therapy
* Any other active malignancies that required systemic treatment within 3 years prior to enrollment
* Radiation therapy within 2 weeks prior to ALLO-647
* Prior irradiation to \>25% of the bone marrow
* Hypocellular bone marrow for age by institutional standard as determined from a bone marrow biopsy performed at time of screening (Note: Only applicable for Phase 2).
* Autologous hematopoietic stem cell transplant (HSCT) within last 6 months (24 weeks)
* Systemic anti-cancer therapy within 2 weeks prior to receiving ALLO-647