Avelumab in Patients With MSS, MSI-H and POLE-mutated Recurrent or Persistent Endometrial Cancer and of Avelumab/Talazoparib and Avelumab/Axitinib in Patients With MSS Recurrent or Persistent Endometrial Cancer

Inclusion Criteria: Participants must be classified into one of the following cohorts of recurrent or persistent endometrial cancer of any histology: The MSI/POLE cohort includes endometrial cancers that are: --MSI-H as determined by immunohistochemical complete loss of expression (absence of nuclear immunoreactivity) of at least one of the mismatch repair genes MSH2, MSH6, MLH1 and PMS2. This test is now done routinely for every newly diagnosed endometrial cancer patient in most centers in the US. And/OR: --POLE-mutated, i.e. endometrial cancers known to harbor mutations in the exonuclease domain (amino acid residues 268-471) of polymerase e (POLE) as determined by targeted sequencing or other next generation sequencing assay. Any Clinical Laboratory Improvement Amendments (CLIA)-approved genomic test documenting mutations in the exonuclease domain of POLE gene (amino acid residues 268-471) in the tumor will be accepted as proof of presence of POLE mutations and will lead to classification into this patient cohort. The MSS cohorts include: * Endometrial cancers that are MSS as determined by normal immunohistochemical nuclear expression of all the mismatch repair genes MSH2, MSH6, MLH1 and PMS2. Tumors which have not been sequenced for POLE mutations (i.e. their POLE mutations status is unknown) but are MSS, will be included in this cohort. * All patients must have measurable disease as defined by RECIST 1.1. Measurable disease is defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded). Each lesion must be \>= 10 mm when measured by CT, MRI or caliper measurement by clinical exam; or \>= 20 mm when measured by chest x-ray. Lymph nodes must be \> 15 mm in short axis when measured by CT or MRI. * Prior Therapy: * There is no upper limit of prior therapies but patients must have had one prior chemotherapeutic regimen for management of endometrial carcinoma. Initial treatment may include chemotherapy, chemotherapy and radiation therapy, and/or consolidation/maintenance therapy. Any platinum based chemotherapy (single agent platinum or any platinum doublet) administered in conjunction with primary radiation as a radio-sensitizer WILL be counted as a systemic chemotherapy regimen. Furthermore, patients who have only received chemotherapy in the adjuvant setting will be eligible for the study. * Prior hormonal therapy is allowed. * Patients must NOT have received any class of drugs targeted to the PD-1/PD-L1 pathway. * Patients must NOT have received any prior PARP inhibitor therapy (for patients being considered for the avelumab/talazoparib cohort only). * Patients must NOT have received prior axitinib (for patients being considered for the avelumab/axitinib cohort only). * Age of 18 or greater years. Because insufficient dosing or adverse event data are currently available on the use of Avelumab, talazoparib, and/or axitinib in participants \< 18 years of age, children are excluded from the study. Endometrial cancer is very rare in the pediatric population. * ECOG performance status 0 or 1 (reference Appendix A for ECOG performance status criteria). * Availability of a formalin fixed paraffin embedded (FFPE) block of cancer tissue OR 15 unstained 5-micron slides from the original surgery or biopsy or from a biopsy of recurrent disease. * Participants must have normal organ and marrow function as defined below: * absolute neutrophil count \>1,500/mcL * platelets \>100,000/mcL * hemoglobin ≥ 9g/dL * total bilirubin within normal institutional limits * AST(SGOT)/ALT(SGPT) ≤2.5 × institutional upper limit of normal * creatinine within normal institutional limits OR * creatinine clearance ≥60 mL/min/1.73 m2 for participants with creatinine levels above institutional normal. Please note: creatinine clearance (CLCR) should be estimated according to the Cockcroft-Gault formula as: CLCR={\[(140-age) × weight)\]/(72 x SCR)} × 0.85 where CLCR (creatinine clearance) is measured in mL/min, age is expressed in years, weight in kilograms (kg), and SCR (serum creatinine) in mg/dL. NOTE: Patients with moderate renal impairment (defined as an estimated creatinine clearance of 30-59 mL/min) will receive a reduced starting dose of Talazoparib at 0.75 mg PO QD. * Participant must not be pregnant or breastfeeding given that avelumab is an agent with unknown effects in pregnancy and breastfeeding and the potential for teratogenesis. Females of childbearing potential are defined as those who are not surgically sterile (i.e., bilateral tubal ligation, bilateral oophorectomy, or complete hysterectomy) or post-menopausal (defined as ≥ 12 months with no menses without an alternative medical cause). Serum pregnancy test (for females of childbearing potential) negative at screening. * The effects of avelumab on the developing human fetus are unknown. For this reason and because some immunomodulatory agents are known to be teratogenic, women of child-bearing potential must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. * Toxicities of prior therapy (excepting alopecia and sensory neuropathy) should be resolved to \< grade 2 per the revised NCI Common Terminology Criteria for Adverse Events (CTCAE) version 4. All appropriate treatment areas should have access to a copy of the CTCAE version 4. A copy of the CTCAE version 4 can be downloaded from the CTEP website at: http://ctep.cancer.gov. * Ability to understand and the willingness to sign a written informed consent document. Additional inclusion criteria for the avelumab/axitinib cohort: * Participants must have adequately controlled blood pressure (BP) with or without antihypertensive medications, defined as systolic BP that must be ≤140 mmHg and diastolic BP that must be ≤90 mmHg on two separate BP readings taken at least 1 hour apart at screening. * Participants must have LVEF ≥ lower limit of normal (LLN) as assessed by either multigated acquisition (MUGA) scan or echocardiogram (ECHO). Exclusion Criteria: * Participants who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from adverse events due to agents administered more than 4 weeks earlier. * Participants who are receiving any other investigational agents. * Participants with known brain metastases should be excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events. * History of allergic reactions attributed to avelumab or any component in its formulations, or compounds of similar chemical or biologic composition to avelumab. Known severe hypersensitivity reactions to monoclonal antibodies (Grade ≥ 3 NCI CTCAE v 4.03), any history of anaphylaxis, or uncontrolled asthma (that is, 3 or more features of partially controlled asthma) * Participants with a history of treatment with an anti-PD-1, anti-PD-L1, anti-CTLA-4 or other investigational agents that target immune checkpoint inhibitors. * Known human immunodeficiency virus (HIV) or acquired immunodeficiency syndrome (AIDS) related illness, which may compromise the efficacy of immunostimulatory therapy. * Positive test for HBV surface antigen * Positive Hepatitis C antibody and positive confirmatory HCV RNA test. The confirmatory HCV RNA test is not required if the HCV antibody is negative. If Hepatitis C antibody is positive, the confirmatory HCV RNA test should be done and if it is negative, then participants are eligible. * Subjects requiring hormone replacement with corticosteroids are eligible if the steroids are administered only for the purpose of hormonal replacement and at doses ≤ 10 mg or 10 mg equivalent prednisone per day * Active infection requiring systemic therapy. * Current or prior use of immunosuppressive medication within 7 days prior to enrollment with the following exceptions to this exclusion criterion: * Intranasal, inhaled, topical steroids, or local steroid injections (eg, intra-articular injection); * Systemic corticosteroids at physiologic doses not to exceed 10 mg/day of prednisone or equivalent; * Steroids as premedication for hypersensitivity reactions (eg, CT scan premedication). * Active autoimmune disease that might deteriorate when receiving an immunostimulatory agent. Patients with diabetes type I, vitiligo, psoriasis, hypo- or hyperthyroid disease not requiring immunosuppressive treatment are eligible. * Prior organ transplantation including allogeneic stem-cell transplantation. * Severe gastrointestinal conditions such as clinical or radiological evidence of bowel obstruction within 4 weeks prior to study entry, uncontrolled diarrhea in the last 4 weeks prior to enrollment, or history of inflammatory bowel disease. * Uncontrolled intercurrent illness including, but not limited to symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements. * 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. * Known alcohol or drug abuse. * Individuals with a history of a different malignancy are ineligible except for the following circumstances: Individuals with a history of other malignancies are eligible if they have been disease-free for at least 5 years and are deemed by the investigator to be at low risk for recurrence of that malignancy. Individuals with the following cancers are eligible if diagnosed and treated within the past 5 years: breast cancer in situ, cervical cancer in situ, and basal cell or squamous cell carcinoma of the skin. * All other significant diseases (for example, inflammatory bowel disease, uncontrolled asthma), which, in the opinion of the Investigator, might impair the subject's tolerance of trial treatment. * Any psychiatric condition that would prohibit the understanding or rendering of informed consent * Vaccination within 4 weeks of the first dose of avelumab and while on trial is prohibited except for administration of inactivated vaccines. * Patients may not use natural herbal products or other "folk remedies" while participating in this study. Herbal medications include, but are not limited to St. John's Wort, Kava, ephedra (ma huang), gingko biloba, dehydroepiandrosterone (DHEA), yohimbe, saw palmetto, and ginseng. Additional exclusion criteria for the avelumab/axitinib cohort: * Participants having \>1+ proteinuria on urinalysis or UPCR \>1 will undergo a 24-hour urine collection for quantitative assessment of proteinuria. Participants with urine protein \>1 g/24-hours will be ineligible. * Participants with concern for bowel or serosal involvement will be ineligible, due to the risk of perforation or fistulization with anti-angiogenic agents. * Participants will be ineligible if they have active gastrointestinal bleeding, as evidenced by clinically significant hematemesis, hematochezia, or melena in the past 3 months without evidence of resolution documented by endoscopy or colonoscopy. * Participants will be ineligible if using anticoagulant therapy with oral vitamin K antagonists, novel oral anticoagulants (NOACs), or direct oral anticoagulants (DOACs), inclusive of direct thrombin inhibitors and direct factor Xa inhibitors. Therapeutic use of low molecular weight heparin is allowed. Low dose heparin required for maintenance of patency of central venous access devices are allowed. * Grade ≥3 hemorrhage within 4 weeks preceding Cycle 1 Day 1 treatment. * Ongoing cardiac dysrhythmias of CTCAE Grade≥2, or prolongation of the QTc interval to \>500 msec * Current use or anticipated need for treatment with drugs or foods that are known to be either: * Strong CYP3A4/5 inhibitors, including administration within 10 days prior to Cycle 1 Day 1 treatment, including but not limited to grapefruit juice, grapefruit-related fruits (Seville oranges, pomelos), ketoconazole, miconazole, itraconazole, voriconazole, posaconazole, clarithromycin, telithromycin, indinavir, saquinavir, ritonavir, nelfinavir, amprenavir, fosamprenavir nefazodone, lopinavir, troleandomycin, mibefradil, conivaptan. The topical use of these medications is allowed if systemic absorption is considered minimal. * Strong CYP3A4/5 inducers, including administration within 10 days prior to Cycle 1 Day 1 treatment, including but not limited to phenobarbital, rifampin, phenytoin, carbamazepine, rifabutin, rifapentine, clevidipine, St. John's wort.