Inclusion Criteria:

          -  Patients must have histologic confirmation of MTAP-deficient metastatic urothelial
             carcinoma. MTAP-deficiency must be verified by institutional Clinical Laboratory
             Improvement Act (CLIA)-certified immunohistochemistry (IHC). Histological variants
             such as glandular, squamous, sarcomatoid, micropapillary, plasmacytoid, and small cell
             changes will be allowed for this trial if these tumors are MTAP-deficient.

          -  Patients can be considered for second line of therapy (after chemotherapy or immune
             checkpoint inhibitor with PD-[L]1 agent) or for third line of therapy (can have
             previously received chemotherapy and immune checkpoint inhibitor with PD-[L]1
             blockade). Any prior intravesical therapy is allowed and does not count as a prior
             line of therapy.

          -  Patients who received methotrexate-containing chemotherapy (e.g.
             methotrexate/vinblastine/adriamycin/cisplatin [MVAC]) as neoadjuvant therapy or
             first-line systemic therapy at least 12 months prior will be allowed for this trial.

          -  All patients must have measurable disease by Response Evaluation Criteria in Solid
             Tumors (RECIST) version (v)1.1 and tumors of sufficient sizes for biopsy. In general,
             liver and lung lesions should be at least 1.0 cm, and patients with lymph node-only
             disease should have lesions of >= 1.5 cm in shortest dimension. Patients with disease
             confined to bone may be eligible if a measurable lytic defect is present. The study
             principal investigator (PI) is the final arbiter in questions related to
             measurability. Patients with a three-dimensional mass or pelvic sidewall fixation on
             bladder examination under anesthesia are considered to have measurable disease.

          -  Patients must have an Eastern Cooperative Oncology Group (ECOG) performance status =<
             2.

          -  Aspartate aminotransferase (AST) or alanine aminotransferase (ALT) =< 2.5 x upper
             limit of normal (ULN), or =< 5 ULN if documented liver metastases are present.

          -  Total bilirubin =< 1.5 x ULN, except subjects with Gilbert's syndrome or liver
             metastases, who must have a baseline total bilirubin =< 3.0 mg/dL.

          -  Absolute neutrophil count (ANC) >= 1.5 x 10^9/L.

          -  Hemoglobin >= 9 g/dL (may have been transfused).

          -  Platelets >= 100 x 10^9/L.

          -  Normal serum creatinine, or a creatinine clearance >= 40 ml/min [either measured using
             a 24 hour urine, calculated using Cockcroft-Gault, or estimated using the MDRD method
             from the National Kidney Disease Education Program (NKDEP) (the method reported by MD
             Anderson Cancer Center [MDACC] laboratories).

          -  Negative serum or urine pregnancy test at screening for women of child-bearing
             potential.

          -  Females of childbearing potential who are sexually active with a non-sterilized male
             partner and non-sterilized males must use a highly effective method of contraception
             for 28 days prior to the first dose of investigational product, and must agree to
             continue using such precautions for 180 days after the final dose of investigational
             product; cessation of contraception after this point should be discussed with a
             responsible physician. They must also refrain from egg cell donation for 180 days
             after the final dose of investigational product.

          -  Non-sterilized males who are sexually active with a female partner of childbearing
             potential must use a highly effective method of contraception from days 1 through 180
             post last dose. In addition, they must refrain from sperm donation for 180 days after
             the final dose of investigational product.

          -  The ability to interrupt nonsteroidal anti-inflammatory drugs (NSAIDS) 2 days before
             (5 days for long-acting NSAIDs), the day of, and 2 days following administration of
             pemetrexed.

          -  The ability to take folic acid, vitamin B12, and dexamethasone according to protocol.

          -  Mild autoimmune conditions (such as localized psoriasis) requiring minimal treatment
             or systemic autoimmune conditions well controlled by target agents such as an
             anti-IL-17 that do not affect overall immune system. Patients with a history of
             Hashimoto's thyroiditis only requiring hormone replacement, type I diabetes, or
             conditions not expected to recur in the absence of an external trigger are allowed to
             participate.

        Exclusion Criteria:

          -  Primary central nervous system (CNS) malignancies or CNS metastases, including
             leptomeningeal metastases, are not allowed. Subjects with previously treated brain
             metastases will be allowed if the brain metastases have been stable for at least 4
             weeks following prior treatment and no ongoing steroid requirement.

          -  Any other malignancy from which the patient has been disease-free for less than 2
             years, except for non-melanomatous skin cancer, controlled localized prostate cancer,
             in situ carcinoma of any site.

          -  Women who are pregnant or breastfeeding or intend to become pregnant during their
             participation in the study.

          -  Presence of third space fluid which cannot be controlled by drainage. For patients who
             develop or have baseline clinically significant pleural or peritoneal effusions (on
             the basis of symptoms or clinical examination) before or during initiation of
             pemetrexed therapy, consideration should be given to draining the effusion prior to
             dosing. However, if, in the investigator's opinion, the effusion represents
             progression of disease, the patient should be discontinued from study therapy.

          -  Known or suspected autoimmune disease. Patients with a history of inflammatory bowel
             disease (including Crohn's disease and ulcerative colitis) and autoimmune disorders
             such as rheumatoid arthritis, systemic progressive sclerosis (scleroderma), systemic
             lupus erythematosus or autoimmune vasculitis (e.g., Wegener's granulomatosis) are
             excluded from this study.

          -  Any condition requiring systemic treatment with corticosteroids (> 10 mg daily
             prednisone equivalents) or other immunosuppressive medications within 14 days prior to
             first dose of study drug. Inhaled steroids and adrenal replacement steroids doses > 10
             mg daily prednisone equivalents are permitted in the absence of active autoimmune
             disease.

          -  History of primary immunodeficiency.

          -  Patients who have prior organ transplantation, including allogeneic stem-cell
             transplant.

          -  Vaccinations within 4 weeks of the first dose of avelumab and while on trials is
             prohibited except for administration of inactivated vaccines.

          -  True positive test results for hepatitis A, B, or C during screening.

          -  Known history of testing positive for human immunodeficiency virus (HIV) or known
             acquired immunodeficiency syndrome (AIDS).

          -  Uncontrolled intercurrent illness including, but not limited to, ongoing or active
             infection, current pneumonitis, symptomatic congestive heart failure, unstable angina
             pectoris, symptomatic cardiac arrhythmia, or interstitial lung disease.

          -  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.

          -  Persisting toxicity related to prior therapy (National Cancer Institute [NCI] Common
             Terminology Criteria for Adverse Events [CTCAE] version [v.] 5.0 grade > 1); however,
             alopecia, sensory neuropathy grade =< 2, or other grade =< 2 not constituting a safety
             risk based on investigator's judgment are acceptable.

          -  Uncontrolled psychiatric illness/social situations that would limit compliance with
             study requirements or compromise the ability of the subject to give written informed
             consent.

          -  Known allergy or hypersensitivity to study drug formulations.

          -  Major surgical procedure (as defined by the PI or co-PIs within 28 days prior to the
             first dose of therapy) or still recovering from prior surgery.

          -  Patient currently on dialysis.