∙ Cohort 1:

• Patient with myeloma who has received at least four prior lines of treatment having been exposed to an IMID, PI, and a CD38 monoclonal antibody and had measurable disease prior to salvage high dose melphalan autoHCT done within the prior 2 - 6 months. (Salvage melphalan/AutoHCT can count as the 4th line of treatment).

• Measurable disease is defined by any of the following:

‣ M-spike ≥ 0.5mg/dL

⁃ Urine m-spike ≥ 200mg/dL/24 hours

⁃ Involved Serum Free light chain ≥ 10mg/dL

⁃ Measurable plasmacytoma on imaging (≥ 1 lesion that has a single diameter ≥ 2 cm).

⁃ Bone marrow plasma cells ≥ 30% as determined by CD138 immunohistochemistry staining

∙ Cohort 2:

• Patients with pathologically confirmed MM who have received at least 4 prior lines of treatment having been exposed to an IMID, PI, and a CD38 monoclonal antibody and have undergone an allo HCT for RRMM at any time in their history and have at least minimal residual disease by flow or NGS in the bone marrow at least 3 months after allo HCT.

• Prior to Leukapheresis:

‣ Greater than age 18.

⁃ Karnofsky performance ≥ 70.

⁃ Recovered to Grade 1 or baseline of any non-hematologic toxicities due to prior treatments, excluding Grade 2 neuropathy

⁃ Not receive any systemic anti-myeloma therapy for 14 days prior to leukapheresis. Therapeutic doses of corticosteroids (defined as greater than 10 mg/day prednisone or equivalent) are permitted until within 72 hours prior to Leukapheresis.

⁃ Absolute neutrophil count (ANC) ≥ 1,000/mm\^3 without filgrastim use in the prior 14 days.

⁃ Hemoglobin ≥ 8 g/dL (without red blood cell transfusion in the previous 7 days)

⁃ Creatinine Clearance (CrCl) ≥ 45 mL/min, measured or estimated by Cockcroft-Gault equation.

⁃ Corrected serum calcium ≤ 13.5 mg/dL

⁃ Oxygen saturation ≥ 92% on room air

• Hepatic Function:

‣ Serum aspartate aminotransferase (AST) or alanine aminotransferase (ALT) ≤ 2.5 x upper limit of normal (ULN

⁃ Serum total bilirubin Serum total bilirubin ≤ 2 x ULN. Patients who have been diagnosed with Gilbert's disease are permitted to exceed the defined bilirubin value of 2 x ULN 2 x ULN. Patients who have been diagnosed with Gilbert's disease are permitted to exceed the defined bilirubin value of 2 x ULN

⁃ International ratio (INR) or partial thromboplastin time (PTT) ≤ 1.5 x ULN

⁃ Cardiac Function: left ventricular ejection fraction ≥ 45% by echocardiogram (ECHO) or multigated acquisition scan (MUGA).

⁃ Willing and able to adhere to the study visit schedule and other protocol requirements including regulatory requirement of a 15 year follow up using the CIBMTR long term follow up mechanism.

• Female patients of childbearing potential (FCBP) must:

‣ Have a negative serum pregnancy test with a sensitivity of at least 50 mIU/mL prior to enrollment

⁃ Agree to use, and be able to comply with, TWO acceptable methods of birth control, one highly effective method and one additional effective (barrier) method AT THE SAME TIME, from screening through at least 1 year following Ide-Cel infusion

⁃ Agree to abstain from breastfeeding from screening through at least 1 year following Ide-Cel infusion

• Male patients must:

‣ Agree to use a condom during sexual contact with a pregnant female or a FCBP, even if he has undergone a successful vasectomy, from screening through at least 1 year following Ide-Cel infusion

⁃ Must not donate sperm from screening through at least 1 year following Ide-Cel infusion

• Prior to LD Chemotherapy

‣ Females of childbearing potential must have a negative serum pregnancy test ≤ 7 days prior to LD chemotherapy

⁃ Platelet count ≥ 50,000/mm\^3 (transfusion allowed)

⁃ ANC ≥ 1,000/mm\^3 (without filgrastim within 72 hours)

• Hepatic Function:

‣ Serum AST and ALT ≤ 2.5 × ULN

⁃ Serum total bilirubin ≤ 2 × ULN. Patients who have been diagnosed with Gilbert's disease are permitted to exceed the defined bilirubin value of 2 x ULN

• CrCl ≥ 45 mL/min, measured or estimated by Cockcroft-Gault equation

• INR or PTT ≤ 1.5 x ULN

• No history of ≥ Grade 2 hemorrhage within 30 days

• No presence of active/uncontrolled infection requiring systemic therapy. Prophylactic antimicrobials are allowed.

• No intercurrent illness or toxicity that would place the subject at undue risk of proceeding to LD chemotherapy

• Must not be taking therapeutic doses of corticosteroids (defined as greater than 10 mg/day prednisone or equivalent) within 72 hours prior to LD chemotherapy. Physiologic replacement, topical, intranasal and inhaled steroids are permitted. Patients on calcineurin inhibitors (cyclosporine or tacrolimus) should have levels considered undetectable per institutional criteria

• No active urinary outflow obstruction

• Availability of manufactured cells

• Patients not meeting these criteria may still be eligible to initiate LD chemotherapy with the approval of the Protocol PI (Principal Investigator).

• Prior to Ide-Cel or Cilta-Cel infusion

• Subjects who meet at least one of the following criteria on the day of scheduled CAR T cell infusion should have its administration delayed:

‣ Suspected or active systemic infection

⁃ Onset of fever ≥ 38°C

⁃ Requirement for supplemental oxygen to keep saturation greater than 91%

⁃ Cardiac arrhythmia not controlled with medical management

⁃ Hypotension requiring vasopressor support

⁃ New onset or worsening of other non-hematologic organ dysfunction ≥ Grade 3

⁃ Taking any of the prohibited medications as described in Section 15

⁃ Significant worsening in clinical status compared to initial eligibility criteria that would, in the opinion of the treating physician, increase the risk of adverse events associated with Ide-Cel or Cilta-Cel infusion.