A Study of Escalating Doses of ASG-22CE Given as Monotherapy in Subjects With Metastatic Urothelial Cancer and Other Malignant Solid Tumors That Express Nectin-4

Inclusion Criteria: * Dose Escalation, Renal insufficiency and CPI-Treated Expansion cohorts: Histologically confirmed Transitional Cell Carcinoma of the Urothelium (TCCU) (i.e., cancer of the bladder, renal pelvis, ureter, or urethra). Subjects with Urothelial Carcinoma with squamous differentiation or mixed cell types are eligible. * Ovarian Expansion Cohort: Subjects with recurrent disease or histologically or cytologically confirmed Stage III/IV diagnosis of epithelial ovarian cancer, fallopian tube cancer, or primary peritoneal carcinoma who have previously progressed while receiving or within 6 months of completing a platinum-containing regimen. * NSCLC Expansion Cohort: Histologic or cytologic diagnosis of NSCLC (squamous or non-squamous or NSCLC-not specified) * Dose Escalation, Renal insufficiency, NSCLC and CPI-Treated Expansion Cohorts: For subjects with urothelial cancer and NSCLC: Subjects must submit a tumor tissue for Nectin-4 expression. Enrollment for these subjects is not dependent on the immunohistochemistry using the H-Score (IHC H-Score). * Ovarian Expansion Cohort: Subjects must have tumor tissue positive (IHC H-score ≥150) for Nectin-4 expression * For Dose Escalation, NSCLC and Ovarian Expansion Cohorts: Subject must have failed at least one prior chemotherapy regimen for metastatic disease (urothelial and bladder cancer subjects are not required to have failed prior chemotherapy regimen if considered unfit for cisplatin-based chemotherapy) * For the CPI-Treated Expansion Cohort: Subject must have received prior treatment with a CPI in the metastatic setting. * Subjects must have measurable disease according to RECIST (version 1.1) * Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 * Life expectancy of ≥ 3 months * Negative pregnancy test (women of childbearing potential) * Hematologic function, as follows (no red blood cell or platelet transfusions are allowed within 14 days of the first dose of enfortumab vedotin): * Absolute neutrophil count (ANC) ≥ 1.0 x 10\^9/L * Platelet count ≥ 100 x 10\^9/L * Hemoglobin ≥ 9 g/dL * Renal function, as follows: Dose Escalation, NSCLC, Ovarian, and CPI Treated Expansion Cohorts: creatinine clearance of ≥ 30 mL/min by the Cockcroft-Gault equation or as measured by 24 hour urine collection. For the Renal Insufficiency Expansion Cohort: creatinine clearance estimate ≥15 ml/min and \<30 ml/min by Cockcroft-Gault equation or as measured by 24 hour urine collection. * Total bilirubin ≤ 1.5 x ULN (upper limit of normal) * Serum albumin ≥2.5 g/dL * Aspartate aminotransferase (AST) ≤ 1.5 x ULN * Alanine aminotransferase (ALT) ≤ 1.5 x ULN * International normal ratio (INR) \< 1.3 or ≤ institutional ULN (or ≤ 3.0 if on therapeutic anticoagulation) * Sexually active fertile subjects, and their partners, must agree to use medically accepted double-barrier methods of contraception (e.g., barrier methods, including male condom, female condom, or diaphragm with spermicidal gel) during the study and at least 6 months after termination of study therapy * Competent to comprehend, sign, and date an independent ethics committee/institutional review board/research ethics board (IEC/IRB/REB) approved informed consent form Exclusion Criteria: * Preexisting sensory neuropathy Grade ≥ 2 * Preexisting motor neuropathy Grade ≥ 2 * Uncontrolled central nervous system metastases * Use of any investigational drug within 14 days prior to the first dose of study drug * Any anticancer therapy within 14 days prior to the first dose of study drug, including: small molecules, immunotherapy, chemotherapy, monoclonal antibody therapy, radiotherapy or any other agents to treat cancer (anti-hormonal therapy given as adjuvant therapy for early-stage estrogen receptor (ER) positive breast cancer is not considered cancer therapy for the purpose of this protocol) * Subjects with immunotherapy related adverse events requiring high doses of steroids (≥ 40 mg/day of prednisone) are not eligible. * Any P-glycoprotein (P-gp) inducers/inhibitors or strong cytochrome P4503A (CYP3A) inhibitors within 14 days prior to the first dose of study drug * Thromboembolic events and/or bleeding disorders ≤ 14 days (e.g., deep vein thrombosis (DVT) or pulmonary embolism (PE)) prior to the first dose of study drug * Documented history of a cerebral vascular event (stroke or transient ischemic attack), unstable angina, myocardial infarction, or cardiac symptoms (including congestive heart failure) consistent with New York Heart Association Class III-IV within 6 months prior to the first dose of enfortumab vedotin. * Known Human Immunodeficiency Virus (HIV) or Acquired Immune Deficiency Syndrome (AIDS) * Subjects with a positive Hepatitis B surface antigen and/or antihepatitis B core antibody. Subjects with a negative polymerase chain reaction (PCR) assay are permitted with appropriate antiviral prophylaxis. * Active Hepatitis C infection. Subjects who have been treated for Hepatitis C infection can be included if they have documented sustained virologic response of ≥ 12 weeks. * Decompensated liver disease as evidenced by clinically significant ascites refractory to diuretic therapy, hepatic encephalopathy, or coagulopathy * Known sensitivity to any of the ingredients of the investigational product enfortumab vedotin (ASG-22CE) * Major surgery within 28 days prior to first dose of study drug * History of another malignancy within 3 years before the first dose of study drug, or any evidence of residual disease from a previously diagnosed malignancy. Subjects with nonmelanoma skin cancer, localized prostate cancer treated with curative intent with no evidence of progression, low-risk or very low risk localized prostate cancer under active surveillance/watchful waiting without intent to treat, or carcinoma in situ of any time (if complete resection was performed) are allowed. * History of uncontrolled diabetes mellitus or diabetic neuropathy within 3 months of the first dose of study drug. Uncontrolled diabetes is defined as hemoglobin A1C (HbA1c) ≥ 8% or HbA1c \> 7 to \< 8% with associated diabetes symptoms (polyuria or polydipsia) that are not otherwise explained. * Currently receiving systemic antimicrobial treatment for active infection (viral, bacterial, or fungal) at the time of first dose of enfortumab vedotin. Routine antimicrobial prophylaxis is permitted. * Condition or situation which may put the subject at significant risk, may confound the study results, or may interfere significantly with subject's participation in the study * Any medical, psychiatric, addictive or other kind of disorder which compromises the ability of the subject to give written informed consent and/or to comply with procedures * Has ocular conditions such as: * Active infection or corneal ulcer (e.g. keratitis) * Monocularity * History of corneal transplantation * Contact lens dependent (if using contact lens, must be able to switch to glasses during the entire study duration) * Uncontrolled glaucoma (topical medications allowed) * Uncontrolled or evolving retinopathy, wet macular degeneration, uveitis, papilledema, or optic disc disorder