Inclusion Criteria:

          -  Biopsy proven metastatic or unresectable renal cell carcinoma with clear cell
             component

          -  Prior treatment with at least 1 VEGFR TKI and 1 PD-1/PD-L1 immune checkpoint inhibitor
             (ICI).Combination VEGFR TKI plus ICI will be counted as 1 line of therapy. During the
             dose escalation portion of the study prior TKI exposure is not required.

          -  Dose escalation portion: No maximum prior lines of therapy. Dose expansion portion:
             maximum of two prior lines of therapy

          -  Adequate Hematologic Function

               -  Absolute Neutrophil Count ≥ 1.5 x 109 / L

               -  Platelet Count ≥ 100 x 10^9 / L

               -  Hemoglobin ≥ 9 g/dL

               -  No transfusion of packed red blood cells or platelets within 21 days of Cycle 1
                  Day 1

          -  Adequate Renal Function ≥ 60 ml/min according to the Cockcroft-Gault formula

             ° Patients with moderate renal impairment (creatinine clearance 30-59 ml/min by
             Cockcroft-Gault) may be eligible in the phase II dose expansion

          -  Adequate Hepatic Function including:

               -  Total bilirubin ≤ 1.5 x upper limit of normal (ULN)

               -  AST ≤ 3 x upper limit of normal (ULN) without liver metastasis

               -  ALT ≤ 3 x upper limit of normal (ULN) without liver metastasis

               -  AST or ALT ≤ 5 x upper limit of normal (ULN) for patients with liver metastasis

               -  Patients with known Gilbert's syndrome may be included if total bilirubin ≤ x 3
                  ULN

          -  Eastern Cooperative Group (ECOG) Performance Status 0-2.

          -  Patients must have measurable disease by RECIST v1.1. At least one measurable lesion
             should not have been previously irradiated.

          -  Women of childbearing potential must have negative urine or serum pregnancy testing at
             screening. All women will be considered childbearing potential unless meeting criteria
             including:

               -  Achieved post-menopausal status as defined by cessation of regular menses for at
                  least 12 consecutive months with no alternative pathological or physiological
                  cause and have follicular stimulation hormone showing postmenopausal state. Women
                  who have been amenorrhoeic for ≥ 12 months are still considered to be of
                  childbearing potential if the amenorrhea is possibly due to prior chemotherapy,
                  anorexia, low body weight, ovarian suppression, anti-estrogen therapy or other
                  medically inducible reasons.

               -  Documented hysterectomy or bilateral oophorectomy surgery

               -  Medically confirmed ovarian failure

               -  Sexually active participants and their partners must agree to use medically
                  accepted methods of contraception (i.e. barrier methods including condoms, female
                  condom, or diaphragm with spermicidal gel) during the study and for 7 months
                  after the last dose of the study treatment for women, and 4 months for men.

          -  Recovery of baseline CTCAE v5.0 grade ≤1 toxicities related to prior study treatments
             unless adverse events are clinically non-significant per investigator's discretion
             and/or stable on supportive therapy if needed.

          -  Patients must be willing and able to comply with trial protocol. This includes
             adhering to the treatment plan, scheduled visits, laboratory and other study
             procedures.

        Participant Inclusion Criteria for Phase II Dose Expansion

          -  Maximum 2 lines of prior therapy. Combination VEGFR TKI plus ICI therapy will be
             accepted as 1 line of therapy.

          -  Adequate Renal Function ≥ 30 ml/min according to the Cockcroft-Gault formula.

               -  Patients with normal renal function (Cr clearance ≥ 60 ml/min by Cockcroft-Gault
                  formula) will receive talazoparib at RP2D.

               -  Patients with moderate renal impairment (Cr clearance 30-59 ml/min by
                  Cockcroft-Gault formula) will receive talazoparib at one dose level lower than
                  RP2D.

        Exclusion Criteria:

          -  Prior treatment with talazoparib or other agents which target PARP

          -  Prior treatment with axitinib. Other VEGFR TKIs are permissible.

          -  Patients < 18 years old

          -  Patients who are pregnant or breast-feeding. Fertile patients who are unwilling or
             unable to use two methods of contraception (at least one of which considered highly
             effective) for duration of study and after study (7 months after last dose of the
             study treatment for women, and 4 months for men)

          -  Prior diagnosis of myelodysplastic syndrome (MDS) or diagnosis of other malignancy
             that requires anti-cancer directed therapy within the last 12 months. Exclusions
             include those cancers that are considered cured by local therapy (e.g. basal cell
             carcinoma, squamous cell carcinoma, ducal carcinoma in situ of breast, bladder of
             cervix) or other cancers that have low malignant potential and do not require systemic
             therapy (i.e. Gleason-grade <6 prostate adenocarcinoma, borderline ovarian malignancy
             / low malignant potential).

          -  Treatment with anti-cancer therapies within 21 days or five half-lives, whichever
             shorter, of start date, including monoclonal antibody, cytotoxic therapy, or another
             investigational agent.

          -  Significant vascular disease (i.e. aortic aneurysm requiring surgical repair, recent
             arterial thrombosis) within 6 months prior to first dose of therapy.

          -  Ejection Fraction (EF) ≤50% by echocardiogram (ECHO). Multi-gated acquisition scan
             (MUGA) should be obtained to estimate EF if quality of ECHO is insufficient.

          -  Uncontrolled hypertension defined as systolic blood pressure (BP) ≥160 mmHg or
             diastolic BP ≥ 100 mmHg despite anti-HTN therapy.

          -  Evidence of bleeding diathesis or significant unexplained coagulopathy (i.e. absent of
             anticoagulation)

          -  Clinical signs or symptoms of gastrointestinal obstruction requirement parenteral
             hydration, parenteral nutrition, or feeding tube.

          -  Uncontrolled effusion management (pleural effusion, pericardial effusion, or ascites)
             which requires recurrent drainage procedures.

          -  Active infection requiring parenteral antibiotic therapy.

          -  History of either positive HCV RNA viral load or anti-HCV antibody screening
             detectable; HBV infection with HBV surface antigen detection and/or positive HBV DNA
             viral load.

          -  Known hypersensitivity to talazoparib or axitinib, or any component in formulations.

          -  Severe acute or chronic medical conditions which may significantly increase the risk
             of study participants, per treating investigator's discretion.

          -  Radiation therapy to any site (including bone) <2 weeks prior to the first dose of
             therapy

          -  Symptomatic brain metastasis or leptomeningeal disease requiring steroid use. Patients
             are eligible if they are neurologically stable for 4 weeks, have completed radiation
             therapy or surgery, and recovered from side effects. Patients must have discontinued
             steroid therapy for at least 2 weeks prior to first dose of study treatment.

          -  Current or anticipated use of potent P-gp inhibitors within 7 days prior to
             randomization or anticipated use during the study

          -  Inability to swallow capsules, known intolerance to talazoparib and axitinib or its
             excipients, known malabsorption syndrome, or other conditions which impair intestinal
             absorption.