A Phase 1/1b, Open-label, Dose-escalation, Dose-expansion, Parallel Assignment Study to Evaluate Safety and Clinical Activity of PBCAR0191 (Azercabtagene Zapreleucel or "Azer-cel") in Subjects With Relapsed/Refractory (r/r) Non-Hodgkin Lymphoma (NHL) and r/r B-cell Acute Lymphoblastic Leukemia (B-ALL)

Key Inclusion Criteria Criteria for B-ALL: • Participant has confirmed unequivocal r/r CD19+ B-ALL. Criteria for NHL and CLL/SLL: • Participant has unequivocal aggressive CD19+ r/r B-cell NHL that is confirmed by tumor biopsy tissue from last relapse after CD19-directed therapy. For Phase 1 Dose Escalation: * Diffuse large B-cell lymphoma (DLBCL) including Richter's transformation * Follicular lymphoma (FL) including Grade 3 or transformed FL * High-grade B-cell lymphoma (HGBCL) * Primary mediastinal lymphoma For Phase 1b Dose Expansion (CAR T-relapsed cohort): * DLBCL not otherwise specified (NOS) * HGBCL * DLBCL transformed from the following indolent lymphoma subtypes (FL, Marginal Zone lymphoma \[MZL\], and Waldenstrom's Macroglobulinemia \[WM\]) * Other large B-cell lymphoma (LBCL) subtypes may be enrolled with approval from the Medical Monitor. * Participants previously treated with CD19-directed autologous CAR T therapies have received no more than 2 lines of therapy after administration of their previous CAR T product. * For the expansion CAR T-relapsed cohort only: Participants must have received autologous CD19-directed CAR T therapy and demonstrated clinical response to the treatment at Day 28 or later, followed by relapse or progression. For Phase 1b dose expansion (CAR T-naive cohort): * DLBCL NOS * DLBCL transformed from the following indolent lymphoma subtypes (FL, MZL, and WM) * HGBCL * FL (Grade 1-3a) * MZL that is fluorodeoxyglucose (FDG)-avid on positron emission tomography (PET) scan * WM * CLL/SLL * Primary central nervous system (CNS) lymphoma (PCNSL) * Other LBCL subtypes may be enrolled with approval from the Medical Monitor. * Participant must have received at least 1-2 prior lines of therapy, depending on histological subtype but no more than 7 systemic lines of anti-cancer therapy. Criteria for both B-ALL, NHL, and CLL/SLL: * Eastern Cooperative Oncology Group performance status score of 0 or 1. * An estimated life expectancy of at least 12 weeks according to the investigator's judgment. * Seronegative for human immunodeficiency virus antibody. * Participant has adequate bone marrow, renal, hepatic, pulmonary, and cardiac function. Key Exclusion Criteria Criteria for B-ALL: • Burkitt cell (L3 ALL) or mixed-lineage acute leukemia. Criteria for NHL: * Requirement for urgent therapy due to tumor mass effects such as bowel obstruction or blood vessel compression. * Active hemolytic anemia. Criteria for B-ALL and NHL: * No active CNS disease, excluding PCNSL * History of another primary malignancy * Any form of primary immunodeficiency (for example, severe combined immunodeficiency disease). * History of hepatitis B or hepatitis C currently receiving ongoing antiviral therapy. Any known uncontrolled cardiovascular disease at the time of Screening that, in the investigator's opinion, renders the participant ineligible * History of hypertension crisis or hypertensive encephalopathy within 3 months prior to Screening. * History of severe immediate hypersensitivity reaction to any of the agents used in this study. * Presence of a CNS disorder that, in the opinion of the investigator, renders the participant ineligible for treatment. * History of concomitant genetic syndrome such as Fanconi anemia, Kostmann syndrome, Shwachman-Diamond syndrome, or any other known bone marrow failure syndrome. * Active uncontrolled autoimmune disease requiring active immunosuppression at the time of Screening (excluding participants needing steroids for physiologic replacement). * Participant has received stem cell transplant within 90 days before Screening. * Participant has active graft-versus-host disease (GvHD) symptoms. * Participant has received a systemic biologic agent for treatment of the disease under study within 28 days of LD, other systemic anti-cancer therapy within 10 days or 5 half-lives (whichever is shorter) of LD, and no pulse steroid for disease control within 3 days of LD. * Radiotherapy within 4 weeks before Screening. * Presence of pleural/peritoneal/pericardial catheter, as well as permeant biliary and ureteral stents (does not apply to intravenous lines). * Participant has received live vaccine within 4 weeks before Screening. Note: Non-live virus vaccines are not excluded. * Participant has received CD19-directed therapy other than autologous CD19-directed CAR T therapy within 90 days of the anticipated start date of LD. Additional criteria apply.