Enfortumab Vedotin With or Without Pembrolizumab in Rare Genitourinary Tumors (E-VIRTUE)

Background: Many cancers of the testicles and urinary tract are rare diseases; these are diseases that affect less than 200,000 people in the United States. It can be hard to study treatments for these diseases. One combination of drugs-enfortumab vedotin (EV) and pembrolizumab-has already been approved to treat some urinary cancers. Researchers want to see if they can help people with other types of testicle and urinary cancers. Objective: To test EV, with or without pembrolizumab, in patients with rarer cancers of the testicles or urinary tract. Eligibility: People aged 18 and older with rarer cancers of the testicles or urinary tract. Design: Participants will be screened. They will have a physical exam with blood and urine tests. Their ability to perform normal daily activities will be tested. They will have exams of their skin and eyes. They will have imaging scans. A biopsy may be needed: A sample of tissue will be removed from the tumor. The study drugs are both given through a tube attached to a needle inserted into a vein in the arm. Some participants will receive treatments 3 times during 28-week cycles; others will receive treatments 2 times during 21-day cycles. All participants may continue to receive treatments for up to 5 years. Imaging scans and other tests will be repeated. Participants who stop taking the drugs will have follow-up visits every 3 to 4 weeks until the disease gets worse. They will have imaging scans and blood tests. After that, follow-up visits will continue by phone every 3 months for up to 5 years after study therapy is finished.

* INCLUSION CRITERIA: * Participants must have histologically confirmed locally advanced or metastatic pure adenocarcinoma of the urinary tract, pure squamous cell carcinoma of the urinary tract, or treatment-refractory testicular germ cell tumors. Note: For the purposes of enrollment, the urinary tract is defined as the renal pelvis, ureter, bladder, and urethra. * Participants must have measurable disease, per RECIST 1.1. * Participants must have locally advanced or metastatic disease defined as new or progressive lesions on cross sectional imaging. * Participants in Cohorts A1, B1, and C1 must have received prior anti-PD-1/PD-L1 therapy in any setting. * Participants in Cohorts A2, B2, and C2 must be immune checkpoint inhibitor naive. * Participants may be systemic treatment naive except for participants with testicular germ cell tumors, who must have received and be refractory to all standard options of curative-intent treatment. * Age \>= 18 years. * Eastern Cooperative Oncology Group (ECOG) performance status \= 70%) * Participants must have adequate organ and marrow function as defined below: * Hemoglobin (Hgb) \>= 9g/dL * Absolute neutrophil count (ANC) \>= 1,500/mcL * Platelets \>= 100,000/mcL * Total bilirubin \