Phase I/II Trial of Cord Blood-Derived NK Cells Genetically Engineered With NY-ESO-1 TCR/IL-15 Cell Receptor for Relapsed/Refractory Multiple Myeloma

Inclusion criteria: 1\. Patients with multiple myeloma with an expression of NY-ESO-1 by immunohistochemistry in the pre-screening or screening tumor sample or PCR NY-ESO-1 testing by Pathology. CD138 by immunostains will be performed to identify plasma cells before testing for NY-ESO-1 2. Patients are HLA-A\*02:01, HLA-A\*2:05, or HLA-A\*2:06 positive on human leukocyte antigen (HLA) typing at any time. 3\. Patients with relapsed or refractory multiple myeloma (MM) (patients with solitary plasmacytoma are not eligible) who meet the following criteria: 1. \> or = 2 prior lines of therapy (including exposure to at least one proteasome inhibitor, immunomodulatory imide drug \[ImiD\], and anti-cd38 antibody and refractory to the last line of therapy) 2. Have measurable disease (serum monoclonal \[M\] protein level ≥ 0.5 g/dL, and/or urine M protein level ≥ 200 mg/24hrs, and/or involved serum free light chain \[FLC\] level ≥10 mg/dL provided the serum-free light-chain ratio is abnormal) \*\* Refractory is defined as a documented progressive disease during or within 60 days (measured from the last dose of any drug within the regimen) of completing treatment with the last anti-myeloma regimen before study entry 4. No anti-myeloma therapy within 7 days of lymphodepleting therapy. Note: Steroids are allowed at any time up until lymphodepletion. Localized radiation for palliation is allowed at any time up until NK cell infusion 5. Prior autologous/allogeneic transplants are allowed. 6. Prior cell therapy is allowed against targets other than NY-ESO-1. 7. Patients must have recovered from systemic toxicity of prior anti-myeloma therapy at the start of lymphodepletion 8. Eastern Cooperative Oncology Group (ECOG) performance status \<= 2 9. Estimated glomerular filtration rate (eGFR using the Chronic Kidney Disease Epidemiology Collaboration \[CKI-EPI\] equation) \>= 30 ml/min/1.73 m\^2 10. Alanine aminotransferase (ALT)/aspartate aminotransferase (AST) =\< 2.5 x upper limit of normal (ULN) or =\< 5 x ULN if documented liver metastases 11. Total bilirubin =\< 1.5 mg/dL, except in subjects with Gilbert's syndrome in whom total bilirubin must be =\< 3.0 mg/dL 12. No history of liver cirrhosis 13. No ascites 14. Cardiac ejection fraction \>= 50% 15. No clinically significant pericardial effusion as determined by an ECHO or MUGA 16. No uncontrolled arrhythmias or symptomatic cardiac disease 17. No clinically significant pleural effusion (per principal investigator \[PI\] discretion) 18. Baseline oxygen saturation \> 92% on room air 19. Able to provide written informed consent 20. 18-80 years of age 21. Weight ≥ 40 kg 22. Absolute neutrophil count (ANC) ≥ 1000 / * Note: Growth factor support is allowed prior to lymphodepletion chemotherapy (LD chemo). Transfusion support is allowed at any time. If cytopenias are related to multiple myeloma, the patient may proceed without meeting above hematologic parameters only if bone marrow plasma cells are \>= 50% 19. Hemoglobin ≥ 8 g/dL * Note: Growth factor support is allowed prior to LD chemo. Transfusion support is allowed at any time. If cytopenias are related to multiple myeloma, the patient may proceed without meeting above hematologic parameters only if bone marrow plasma cells are \>= 50% 20. Platelet count \>= 25,000 /uL * Note: Growth factor support is allowed prior to LD chemo. Transfusion support is allowed at any time. If cytopenias are related to multiple myeloma, the patient may proceed without meeting above hematologic parameters only if bone marrow plasma cells are \>= 50% 21. All participants who are able to have children must practice effective birth control while on study and up to 3 months post completion of study therapy. Acceptable forms of birth control for female patients include: hormonal birth control, intrauterine device, diaphragm with spermicide, condom with spermicide, or abstinence, for the length of the study. If the participant is a female and becomes pregnant or suspects pregnancy, she must immediately notify her doctor. If the participant becomes pregnant during this study, she will be taken off this study. The study team will ask for information about the pregnancy 22. Men who are able to have children must use effective birth control while on the study. If the male participant fathers a child or suspects that he has fathered a child while on the study, he must immediately notify his doctor 23. Signed consent to long-term follow-up protocol PA17-0483 to fulfill the institutional responsibilities to various regulatory agencies 24. Patients with relapsed or refractory plasma cell leukemia who have received at least two previous regimens CRITERIA FOR LYMPHODEPLETION: Patient should continue to meet eligibility criteria above with the following exceptions: \* Platelet count \>/= 25,000 /μL \*\* Note: Growth factor support is allowed prior to LD chemo. Transfusion support is allowed at any time. If cytopenias are related to multiple myeloma, the patient may proceed without meeting above hematologic parameters only if bone marrow plasma cells are \>= 50% CRITERIA FOR CELL INFUSION: Patients who meet one of the following criteria on the day of infusion will have their administration delayed for 24 hours. If these problems persist beyond 24 hours, patients will not receive their cell infusion. 1. Cardiac arrhythmias not controlled with medical management 2. Hypotension requiring vasopressor support 3. Suspected or active uncontrolled infection Exclusion Criteria 1. Active or uncontrolled infection at the start of lymphodepletion and/or cell infusion 2. Patients with concurrent autoimmune diseases with neurologic involvement, such as multiple sclerosis 3. Participants who have received any live vaccines within 30 days prior to study entry 4. Any active infection requiring systematic antibiotics 5. Any evidence of another malignancy within the last 2 years prior to screening that has not been treated with curative intent (except in situ non-melanoma skin cell cancers and/or carcinoma in-situ of the cervix or other conditions that are deemed low-risk after discussion with the medical monitor) 6. Any major surgery within 28 days of lymphodepletion, minor surgery within 14 days of lymphodepletion, or any planned medical or surgical procedure that in the opinion of the investigator, might jeopardize the patient's safety