A Pilot Study to Investigate the Safety and Clinical Activity of Avelumab (MSB0010718C) in Thymoma and Thymic Carcinoma After Progression on Platinum-Based Chemotherapy

Background: Thymoma and thymic carcinoma are cancers originating in the thymus gland. Platinum-based chemotherapy is standard treatment for them. But not uncommonly, the disease returns and people need more treatment to keep the cancer from growing. The drug Avelumab could help the immune system fight cancer. Objective: To test if avelumab is safe and well-tolerated, and is effective in treating relapsed or refractory thymoma and thymic carcinoma. Eligibility: People ages 18 and older with thymoma or thymic carcinoma that has returned or progressed after platinum-containing chemotherapy Design: Participants will be screened with: * Blood, urine, and heart tests * Scan: They lie in a machine that takes pictures of the body. * Physical exam * Medical history * Biopsy: a needle removes a piece of tumor. Samples can be from a previous procedure, although it is desirable to undergo a new biopsy. Participants will have treatment in 2-week cycles. They will continue until the side effects are not tolerable or their disease gets worse. Visits at the following time points are required per protocol. Patients who respond to treatment or have durable stability after at least 12 months of therapy may undergo a dose de-escalation regimen to continue on therapy. * Every 2 weeks: Participants will get avelumab by infusion in a vein (IV). They will get diphenhydramine (benadryl) and acetaminophen (tylenol) by mouth or IV before receiving avelumab to decrease the chances of developing a reaction to avelumab. They will have blood, urine, and heart tests periodically. * Cycles 4 and 7, then every 6 weeks: Scans will be performed to look for shrinkage or growth of tumor. * Cycle 4: Participants will be offered a chance to undergo a biopsy. * 2-4 weeks after stopping treatment: Blood, urine, and heart tests will be performed. Participants might undergo a scan. * 10 weeks after stopping treatment: Blood, urine, and heart tests. * About 6 months after stopping treatment, then every 3 months: Participants will have scans andcan allow genetic testing on their blood and tissue samples.

* INCLUSION CRITERIA: * Participants must have histologically confirmed thymoma or thymic carcinoma by the pathology department/CCR/NCI. * Participants must have had at least one prior line of platinum-based chemotherapy or participant must have refused cytotoxic chemotherapy. Progressive disease must be documented prior to study entry and participants must have advanced, unresectable disease that is not amenable to surgical resection. * Prior treatment of PD-1 or PD-L1-directed immune checkpoint blockade is permitted if treatment was not discontinued due to disease progression or lifethreatening adverse events per the investigators discretion (laboratory abnormalities alone with prior therapy will not exclude participants from this trial). * Disease must be measurable with at least 1 unidimensional measurable lesion by RECIST 1.1. * Individuals aged greater than or equal to 18 years \-- Because no dosing or adverse event data are currently available on the use of Avelumab in participants \ 1) however, alopecia, sensory neuropathy Grade less than or equal to 2, or other Grade less than or equal to 2 not constituting a safety risk based on investigator s judgment are acceptable. * Pregnancy or lactation period. Note: a negative pregnancy test is required for individuals of childbearing potential. Individuals who are postmenopausal (age-related amenorrhea greater than or equal to 12 consecutive months or follicle-stimulating hormone (FSH) \> 40 milli international units per milliliter \[mIU/ml\]), or who had undergone hysterectomy or bilateral oophorectomy are exempt from pregnancy testing. If necessary to confirm postmenopausal status a FSH level will be included at screening. * Pregnant individuals are excluded from this study because Avelumab is in the class of agents known as antineoplastics/monoclonal antibodies with the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with Avelumab, breastfeeding should be discontinued if the mother is treated with Avelumab. * Known alcohol or drug abuse. * Clinically significant (i.e. active) cardiovascular disease: cerebral vascular accident/stroke (\< 6 months prior to enrollment), myocardial infarction (\< 6 months prior to enrollment), unstable angina, congestive heart failure (\>= New York Heart Association Classification Class II), or serious cardiac arrhythmia requiring medication. * All other severe acute or chronic medical conditions including immune colitis, inflammatory bowel disease, immune pneumonitis, pulmonary fibrosis or psychiatric conditions including recent (within the past year) or active suicidal ideation or behavior; or laboratory abnormalities that may increase the risk associated with study participation or study treatment administration or may interfere with the interpretation of study results and, in the judgment of the investigator, would make the participants inappropriate for entry into this study. * Any psychiatric condition that would prohibit the understanding or rendering of informed consent. * Legal incapacity or limited legal capacity. * Non-oncology vaccine therapies for prevention of infectious disease (e.g. seasonal flu vaccine, human papilloma virus vaccine) within 4 weeks of study drug administration with the exception of vaccinations against COVID-19. Vaccination while on study is also prohibited except for administration of inactivated vaccines (e.g. inactivated influenza vaccines) and vaccinations against COVID-19. * HIV-positive TET participants are ineligible because of the risk of developing opportunistic infections after treatment with an immune checkpoint inhibitor. Prior cases of disseminated herpes virus and fungal infections have been documented in this patient population.