Inclusion Criteria:

          -  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 such as: BRCA1/2, PALB2, RAD51C/D, 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 (Arms A-C)

               -  Defect in MET, ALK or ROS1, e.g. MET mutations or amplifications, high MET
                  expression, ALK translocations, ROS1 translocations (eligible for Arm C:
                  talazoparib + crizotinib)

               -  NOTE: Patients with metastatic castration-resistant prostate cancer can enroll in
                  Arm B with talazoparib + axitinib without a specific and/or selected mutation

               -  Patients who are eligible for more than one Arm will be assigned according to
                  physician preference

          -  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

          -  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 patients at
             time of enrollment even if a previous biopsy is available. Optional on-treatment and
             at-progression biopsies will be encouraged for all patients

          -  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 patients may have bone
             metastatic disease evaluable by Prostate Cancer Working Group 2 (PCWG2) for subjects
             with metastatic castration-resistant prostate cancer (mCRPC), or according to the
             tumor evaluation criteria best suited and accepted for the tumor type being evaluated)

          -  Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0 to 1

          -  Absolute Neutrophil Count (ANC) >= 1,500/mm^3 or >= 1.5 x 10^9/L (without
             hematopoietic growth factor or transfusion support within 14 days prior to study
             enrollment)

          -  Platelets >= 100,000/mm^3 or >= 100 x 10^9/L (without hematopoietic growth factor or
             transfusion support within 14 days prior to study enrollment)

          -  Hemoglobin >= 9 g/dL (>= 5.6 mmol/L) (without hematopoietic growth factor or
             transfusion support within 14 days prior to study enrollment)

          -  estimated creatinine clearance >= 60 mL/min will be required during dose-escalation
             phase, according to the Cockcroft-Gault formula

               -  Where creatinine clearance (CLCR) (creatinine clearance) is measured in mL/min,
                  age is expressed in years, weight in kilograms (kg), and SCR (serum creatinine)
                  in mg/dL

               -  Or as measured by 24h urine assessment 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

          -  Total serum bilirubin =< 1.5 x the upper limit of normal range (ULN)

          -  Aspartate and Alanine aminotransferase (aspartate aminotransferase [AST] and alanine
             aminotransferase [ALT]) =< 5 x ULN

          -  Female Patients of childbearing potential must have negative serum pregnancy or urine
             pregnancy test at screening. Female patients 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 patients are considered
                  to be of childbearing potential

          -  Evidence of a personally signed and dated informed consent document, within > 28 days
             prior to enrollment, indicating that the patient has been informed of all pertinent
             aspects of the study

          -  Willing and able to comply with scheduled visits, treatment plan, laboratory tests,
             and other study procedures

          -  Able to swallow the study drug, have no known intolerance to study drugs or
             excipients, and comply with study requirements

        Exclusion Criteria:

          -  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)

          -  Major surgery within 4 weeks prior to study enrollment

          -  Patients 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)

          -  Diagnosis of myelodysplastic syndrome (MDS)

          -  Known symptomatic brain metastases requiring steroids. Patients 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, patients who required a single
             dose of corticosteroids on days receiving radiation treatment do not require a 2-week
             washout

          -  Participation in other studies involving investigational drug(s) within 4 weeks prior
             to study entry and/or during study participation

          -  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 and sensory neuropathy Grade =< 2, or other Grade =< 2 adverse events not
             constituting a safety risk, based on the investigator's judgement, are acceptable

          -  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

          -  Patients with known uncontrolled human immunodeficiency virus (HIV) virus or acquired
             immunodeficiency syndrome. Note: Patients with history of controlled HIV virus will be
             considered eligible for this trial

          -  Patients with uncontrolled hepatitis B virus (HBV) or hepatitis C virus (HCV)
             infection at screening. Note: Patients with controlled hepatitis B or hepatitis C will
             be considered eligible for this trial

          -  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

               -  Uncontrolled hypertension as indicated by systolic blood pressure > 170 mm Hg or
                  diastolic blood pressure > 105 mm Hg at screening

          -  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: Patients who have recently been on enzalutamide require a 28 day washout
                  period due to longer elimination half-life of this therapy

          -  Patients 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

          -  Inability to swallow capsules, known malabsorption syndrome, or other conditions that
             may impair absorption of study drugs

          -  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

          -  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 patient inappropriate for entry into
             this study

          -  Medical, psychological, or social conditions that may interfere with the patient's
             participation in the study, or with the evaluation of the study results

          -  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

          -  Pregnant female patients; breastfeeding female patients; fertile male patients; and
             female patients 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 patients or 4 months after the last dose of study
             drugs for male patients, whichever is later for the individual patient. Male patients
             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 patient 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
                  postvasectomy ejaculate

               -  Bilateral tubal ligation or bilateral oophorectomy