Open-label Dose-finding and Dose-expansion Study to Evaluate the Safety, Expansion, Persistence, and Clinical Activity of UCART20x22 in Subjects With Relapsed or Refractory B-Cell Non-Hodgkin Lymphoma (B-NHL)

Inclusion Criteria: * Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 * Relapsed or refractory (R/R) mature B-NHL per 2016 WHO criteria and positive for CD20 and/or CD22 * Subjects with NHL subtypes defined by WHO: * Dose-Finding Part: R/R mature B-NHL (except chronic lymphocytic leukemia/small lymphocytic leukemia \[CLL/SLL\], Richter's transformation from prior CLL/SLL, Burkitt's lymphoma, and Waldenstrom's macroglobulinemia) * Dose-Expansion Part: R/R LBCL, defined as: i. DLBCL; ii. High-grade B-cell lymphoma with MYC and BCL2 and/or BCL6 rearrangements; iii. Transformed FL or transformed marginal zone lymphoma (MZL); iv. Follicular lymphoma Grade 3B * R/R disease after at least 2 lines of prior treatment, which must have included: * An Anti-CD20 MoAb and an anthracycline for DLBCL, high-grade B-cell lymphoma with MYC and BCL2 and/or BCL6 rearrangements, primary mediastinal large B-cell lymphoma (PMBCL), or transformed FL or MZL * An alkylating agent in combination with an anti-CD20 MoAb for FL * An anthracycline or bendamustine-containing chemotherapy regimen and a Bruton's tyrosine kinase (BTK) inhibitor for mantle cell lymphoma (MCL) * Autologous anti-CD19 CAR T-cell therapy, if approved and available for the indicated lymphoma subtype, unless the subject is unable or is ineligible to receive approved autologous anti-CD19 CAR T-cell therapy (e.g., fail leukapheresis or manufacture, unable to wait for manufacture, CD19 negative disease, etc.) * Autologous hematopoietic stem cells must be available prior to the start of the LD regimen if the subject is considered high-risk for prolonged hematologic toxicity. Exclusion Criteria: * Prior use of an investigational product (except for cell or gene therapies and MoAbs) within 5 half-lives or within 14 days, whichever is shorter, prior to start of LD regimen * Previous approved therapy including chemotherapy, biologic (except MoAbs), or targeted therapy for R/R B-NHL with 5 half-lives or within 14 days, whichever is shorter, prior to start of the LD regimen * \> 4 lines of therapy R/R B-NHL prior to start of the LD regimen. * Prior MoAb therapy (approved or investigational) within 30 days prior to start of LD * Prior systemic immunostimulatory agent within 3 half-lives prior to start of the LD regimen * Prior cell or gene therapy (approved or investigational) within 6 months of the start of LD * Prior cell or gene therapy (approved or investigational) targeting both CD20 and CD22 * Autologous HSCT infusion within 6 weeks of the start of LD * Allogeneic HSCT within 3 months of the start of LD, or donor lymphocyte infusion within 6 weeks of the start of LD * Active acute or chronic graft versus host disease (GvHD). Subjects should be off all immunosuppressive therapies for at least 6 weeks prior to start of LD * Radiotherapy within 8 weeks (except for palliative radiotherapy for specific on-target lesions) (prior to start of LD regimen) * Evidence of active central nervous system (CNS) lymphoma or previous CNS involvement of R/R B-NHL * Presence of an active and clinically relevant CNS disorder * Daily treatment with \>20 mg prednisone or equivalent * Known active infection, or reactivation of a latent infection, whether bacterial or viral, fungal, mycobacterial, or other pathogens * History of hypersensitivity to alemtuzumab * History of neutralizing anti-drug antibody against alemtuzumab * Any known uncontrolled cardiovascular disease within 3 months of enrollment * Subjects requiring immunosuppressive treatment * Major surgery within 28 days prior to start of LD * Evidence of another uncontrolled malignancy within 2 years prior to Screening (except in situ nonmelanoma skin cell cancers and/or carcinoma in-situ of the cervix)