Talazoparib and Palbociclib, Axitinib, or Crizotinib for the Treatment of Advanced or Metastatic Solid Tumors, TalaCom Trial

Inclusion Criteria: 1. Pathogenic or likely pathogenic germline or somatic gene defect as determined by local assessment and classification in at least one of the following: * Defect in DNA Damage Response (DDR) genes specified below for each cohort or other related genes at the discretion of the principal investigator in consultation with the MD Anderson Cancer Center Institute for Personalized Cancer Therapy Precision Oncology Decision Support (PODS) group. See below for additional eligibility guidance for Arms A - C: * Eligibility for Arm A (Talazoparib + Palbociclib): * Solid tumors with defects in DDR genes such as: BRCA1/2, PALB2, RAD51C/D, or other related genes at the discretion of the principal investigator, or * MYC-aberrant solid tumors (e.g. overexpression, amplification, mutation) * Eligibility for Arm B (Talazoparib + Axitinib): * Solid tumors with defects in DDR genes such as: BRCA1/2, PALB2, RAD51C/D, or other related genes at the discretion of the principal investigator, or * Participants with BRCA1/2 wild-type high-grade serous ovarian cancer, or * Participants with metastatic castration-resistant prostate cancer without a specific and/or selected mutation * Eligibility for Arm C (Talazoparib + Crizotinib): * Solid tumors with defects in DDR genes such as: BRCA1/2, PALB2,36 RAD51C/D, or other related genes at the discretion of the principal investigator, or * Solid tumors with defects in MET, ALK or ROS1 (e.g. MET mutations or amplifications, high MET expression, ALK translocations, ROS1 translocations) NOTE: Participants who are eligible for more than one Arm will be assigned according to physician preference. 2. Histological or cytological diagnosis of a solid tumor that is advanced/metastatic, intolerable to standard therapy, resistant to effective standard therapy, or for which no standard therapy is available. 3. Availability of a fresh or recent tumor tissue sample from a diagnostic biopsy/surgery or a metastatic tumor biopsy; the sample must have been obtained within 12 months prior to study enrollment. When only bone disease is present, an archival tumor tissue sample obtained within 5 years prior to study enrollment may be accepted for non-prostate cancer patients and a fresh bone biopsy may be accepted for prostate cancer patients only. NOTE: A fresh biopsy should be encouraged for all participants at time of enrollment even if a previous biopsy is available. Optional on-treatment and at-progression biopsies will be encouraged for all participants. With agreement of the PI, if the archived tissue is not available and/ or it is unsafe to obtain a fresh biopsy at screening, the participant is eligible. 4. Have measurable disease at study enrollment as defined by RECIST v1.1 with at least 1 measurable lesion that has not previously been irradiated; or participants may have bone metastatic disease evaluable by Prostate Cancer Working Group 2 (PCWG2) for participants with metastatic castration-resistant prostate cancer (mCRPC), or according to the tumor evaluation criteria best suited and accepted for the tumor type being evaluated). 5. Age . 18 years old. 6. Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0 to 1. 7. Adequate bone marrow function (without hematopoietic growth factor or transfusion support within 14 days prior to study enrollment), including: * Absolute Neutrophil Count (ANC) ≥ 1,500/mm3 or ≥ 1.5 x 109/L * Platelets ≥ 100,000/mm3 or ≥ 100 x 109/L * Hemoglobin ≥ 9 g/dL (≥ 5.6 mmol/L). 8. Adequate renal function defined by an estimated creatinine clearance (CRCL) ≥ 60 mL/min OR normal creatinine as assessed by 24h urine assessment (not both) will be required during the dose-escalation phase. * Calculate CRCL according to the Cockcroft-Gault formula as: CRCL = \[(140-age) × weight)\]/(72 x SCR)\] × 0.85 (if female), where CRCL (creatinine clearance) is measured in mL/min, age is expressed in years, weight in kilograms (kg), and SCR (serum creatinine) in mg/dL 24h urine assessment: * 24h urine assessment will be performed to assess whether creatinine is within normal limits * In the event that 24h urine is used, then CRCL does not require to be calculated NOTE: Patients with moderate renal impairment (30 - 59 mL/min) will be considered during the dose expansion phase. A reduced starting dose for talazoparib will be considered in these patients. If the dose for dose expansion is 1 mg once daily, the dose will be reduced to 0.75 mg once daily. If the expanded dose is 0.75 mg once daily, the dose will be reduced to 0.5 mg once daily. 9. Adequate liver function including: * Total serum bilirubin ≤ 1.5 x the upper limit of normal range (ULN) * Aspartate and Alanine aminotransferase (AST and ALT) ≤ 5 x ULN. 10. Female participants of childbearing potential must have negative serum pregnancy or urine pregnancy test at screening. Female participants of non-childbearing potential must meet at least one of the following criteria: * Achieved postmenopausal status, defined as follows: cessation of regular menses for at least 12 consecutive months with no alternative pathological or physiological cause and have a serum follicle-stimulating hormone (FSH) level confirming the postmenopausal state * Have undergone a documented hysterectomy and/or bilateral oophorectomy * Have medically confirmed ovarian failure All other female participants are considered to be of childbearing potential. 11. Evidence of a personally signed and dated informed consent document, within 28 days prior to enrollment, indicating that the participant has been informed of all pertinent aspects of the study. 12. Willing and able to comply with scheduled visits, treatment plan, laboratory tests, and other study procedures. 13. Able to swallow the study drug, have no known intolerance to study drugs or excipients, and comply with study requirements. Exclusion Criteria: 1. Prior anti-