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-cancer therapy within 2 weeks prior to study enrollment or prior radiation therapy within 2 weeks prior to study enrollment. Prior palliative radiotherapy to metastatic lesion(s) is permitted, provided it has been completed at least 2 days prior to study enrollment and no clinically significant toxicities are expected (e.g. mucositis, esophagitis). 2. Major surgery within 4 weeks prior to study enrollment. 3. Participants with known hypersensitivity to either talazoparib or the additional study drug to be received per treatment arm: palbociclib (Arm A), axitinib (Arm B), crizotinib (Arm C). 4. Diagnosis of myelodysplastic syndrome (MDS). 5. Known symptomatic brain metastases requiring steroids. Participants with previously diagnosed brain metastases are eligible if they have completed their treatment and have recovered from the acute effects of radiation therapy or surgery prior to study enrollment, have discontinued corticosteroid treatment for these metastases for at least 2 weeks, and are neurologically stable. Of note, participants who required a single dose of corticosteroids on days receiving radiation treatment do not require a 2-week washout. 6. Persisting toxicity related to prior therapy (NCI CTCAE v5.0 Grade \> 1). However, alopecia and sensory neuropathy Grade ≤ 2, or other Grade ≤ 2 adverse events not constituting a safety risk, based on the investigators judgement, are acceptable. 7. Active infection requiring systemic therapy. Minor infections, e.g. periodontal infection or urinary tract infection (UTI), which may be treated with short term oral antibiotics are allowed. 8. Participants with known uncontrolled HIV virus or Acquired immunodeficiency syndrome. Note: Patients with history of controlled HIV virus will be considered eligible for this trial. 9. Participants with uncontrolled Hepatitis B virus (HBV) or hepatitis C virus (HCV) infection at screening. Note: Participants with controlled hepatitis B or hepatitis C will be considered eligible for this trial. 10. Clinically significant cardiovascular disease, including any of the following: * Myocardial infarction or symptomatic cardiac ischemia within 6 months before screening; * Congestive heart failure New York Heart Association class III or IV; * History of clinically significant ventricular arrhythmias (eg, sustained ventricular tachycardia, ventricular fibrillation, torsade de pointes) within 1 year before screening; * History of Mobitz II second degree or third degree heart block unless a permanent pacemaker is in place; * Hypotension as indicated by systolic blood pressure \<86 mm Hg at screening; * Bradycardia as indicated by a heart rate of \<45 beats per minute on the screening electrocardiogram; * For part B participants only: Uncontrolled hypertension as indicated by systolic blood pressure \>140 mm Hg or diastolic blood pressure \>90 mm Hg despite optimal treatment. Participants can only be on one antihypertensive and stable doses of antihypertensives at discretion of PI. 11. Current use of potent P-gp inhibitors within 7 days prior to enrollment: amiodarone, carvedilol, clarithromycin, cobicistat, dronedarone, erythromycin, glecaprevir/pibrentasvir, indinavir, itraconazole, ketoconazole, lapatinib, lopinavir, propafenone, quinidine, ranolazine, ritonavir, saquinavir, sofosbuvir/velpatasvir/voxilaprevir, telaprevir, tipranavir, valspodar, and verapamil. • NOTE: Participants who have recently been on enzalutamide require a 28 day washout period due to longer elimination half-life of this therapy. 12. Participants treated within the last 7 days prior to enrollment with: • Food or drugs that are known to be strong CYP (cytochrome P-450) 3A4 inhibitors (ie, amprenavir, atazanavir, boceprevir, clarithromycin, conivaptan, delavirdine, diltiazem, erythromycin, fosamprenavir, indinavir, itraconazole, ketoconazole, lopinavir, mibefradil, miconazole, nefazodone, nelfinavir, posaconazole, ritonavir, saquinavir, telaprevir, telithromycin, verapamil, voriconazole, and grapefruit or grapefruit juice). Drugs that are known to be strong CYP3A4 inducers (ie, carbamazepine, felbamate, nevirapine, phenobarbital, phenytoin, primidone, rifabutin, rifampin, rifapentin, and St. John's wort. 13. Inability to swallow capsules, known malabsorption syndrome, or other conditions that may impair absorption of study drugs. 14. Bisphosphonate or denosumab dosage that was not stable (i.e. not the same) for at least 2 weeks before study enrollment for patients receiving these therapies. 15. Other acute or chronic medical or psychiatric condition including recent (within the past year) or active suicidal ideation or behavior or laboratory abnormality that may increase the risk associated with study participation or investigational product administration or may interfere with the interpretation of study results and, in the judgement of the investigator, would make the participant inappropriate for entry into this study. 16. Medical, psychological, or social conditions that may interfere with the participant's participation in the study, or with the evaluation of the study results. 17. Diagnosis of any other malignancy within 2 years prior to study enrollment, except for adequately treated basal cell or squamous cell skin cancer, or carcinoma in situ of the breast, bladder, or cervix, or low grade (Gleason ≤ 6) prostate cancer on surveillance without any plans for treatment intervention (e.g. surgery, radiation, or castration), or other early-stage low risk cancers. 18. Pregnant female participants; breastfeeding female participants; fertile male participants; and female participants of childbearing potential who are unwilling or unable to use 2 methods of contraception for the duration of the study and for at least 7 months after the last dose of study drugs for female participants or 4 months after the last dose of study drugs for male participants, whichever is later for the individual patient. Male participants are prohibited from sperm donation while enrolled in this study and for 4 months after the last dose of the study drugs. Highly effective methods of contraception are those that alone or in combination, result in a failure rate of less than 1% per year when used consistently and correctly. These methods include: * Established use of oral, inserted, or injected or implanted hormonal methods of contraception are allowed provided the participant remains on the same treatment throughout the entire study and has been using that hormonal contraceptive for an adequate period of time to ensure effectiveness. * Correctly placed copper containing intrauterine device (IUD). * Male condom or female condom used with spermicide (i.e. foam, gel, film, cream or suppository). * Male sterilization with appropriately confirmed absence of sperm in the post-vasectomy ejaculate. * Bilateral tubal ligation or bilateral oophorectomy.