Inclusion Criteria  Patients must be > 12 months and < 30 years at the time of enrollment on study.  Phase I  Patients with refractory or recurrent non-central nervous system (CNS) solid tumors not amenable to curative treatment are eligible. Patients must have had histologic verification of malignancy at original diagnosis or at the time of relapse. Patients eligible for the expansion cohort, A2, will include non-ES patients with refractory or recurrent non-CNS solid tumors with a deleterious alteration in germline or somatic genes involved in HR repair and DSBs signaling, germline or somatic assessed by prior comprehensive sequencing performed in a CLIA-approved (or equivalent) facility.  Phase II  Patients with refractory or recurrent Ewing sarcoma (during or after completion of first-line therapy). Refractory disease is defined as progression during first line treatment or within 12 weeks of completion of first line treatment. Recurrent disease includes patients who received first line treatment and experienced disease progression at any time point >12 weeks from the completion of first line therapy. Patients must have a histologic diagnosis of Ewing sarcoma with EWSR1- FLI1 translocation or other EWS rearrangement at the time of initial diagnosis. Repeat biopsy at the time of disease recurrence is strongly encouraged but it is not required/mandated for enrollment.  Disease status  Patients must have either measurable or evaluable disease (see Section 7.0 for definitions). Measurable disease includes soft tissue disease evaluable by cross-sectional imaging (RECIST). Patients with bone disease without a measurable soft tissue component or bone marrow disease only are eligible for the phase 1 and phase 2 study but will not be included in the OR endpoint. Performance level: Karnofsky > 50% for patients > 16 years of age and Lansky > 50% for patients < 16 years of age. Patients who are unable to walk because of paralysis, but who are up in a wheelchair, will be considered ambulatory for the purpose of assessing the performance score.  Prior therapy  Phase I Patients who have received prior therapy with an irinotecan-based or temozolomide-based regimen are eligible. Patients who have received prior therapy with a PARP inhibitor other than talazoparib are eligible.  Phase II  Patients should have received first line therapy and developed either refractory or recurrent disease (first relapse). Organ function: Must have adequate organ and bone marrow function as defined by the following parameters: Patients with solid tumors not metastatic to bone marrow: Peripheral absolute neutrophil count (ANC) >1,000/mm3 (1x109/L) Platelet count > 75,000/mm3 (75x109/L) (no transfusion within 7 days of enrollment) Hemoglobin > 9 g/dL (with or without support)  In the phase I study, patients with solid tumors metastatic to bone marrow or with bone marrow hypocellularity defined as <30% cellularity in at least one bone marrow site will be eligible for study, but they will not be evaluable for hematologic toxicity. These patients must not be refractory to red cell or platelet transfusions. At least 2 of every cohort of 3 patients (in the phase I study) must be evaluable for hematologic toxicity. If dose limiting hematologic toxicity is observed at any dose level, all subsequent patients enrolled at that dose level must be evaluable for hematologic toxicity.  Adequate renal function defined as: Creatinine clearance or radioisotope GFR > 60ml/min/1.73m2 or a serum creatinine maximum based on age/sex: age 6months to <1 year, creatinine 0.4; 1 to < 2 years, creatinine 0.6; 2 < 6 years, creatinine 0.8; 6 < 10 years, creatinine 1; 10 to <13 years, creatinine 1.2; 13 to < 16 years creatinine 1.5 (males) or 1.4 (females); > 16 years, creatinine 1.7 (males) 1.4 (females) Adequate liver function defined as: normal liver function as defined by SGPT (ALT) concentration <5x the institutional ULN, a total bilirubin concentration <2x the institutional ULN for age, and serum albumin > 2g/dL. Adequate pulmonary function defined as no evidence of dyspnea at rest and a pulse oximetry > 94% if there is a clinical indication for determination. Pulmonary function tests are not required. Patients must have fully recovered from the acute toxic effects of chemotherapy, immunotherapy, surgery, or radiotherapy prior to entering this study: Myelosuppressive chemotherapy: Patient has not received myelosuppressive chemotherapy within 3 weeks of enrollment onto this study (8 weeks if received prior myeloablative therapy). Hematopoietic growth factors: At least 7 days must have elapsed since the completion of therapy with a growth factor. At least 14 days must have elapsed after receiving pegfilgrastim. Biologic (anti-neoplastic agent): At least 7 days must have elapsed since completion of therapy with a biologic agent. For agents that have known adverse events occurring beyond 7 days after administration, this period prior to enrollment must be extended beyond the time during which adverse events are known to occur. Monoclonal antibodies: At least 3 half-lives must have elapsed since prior therapy that included a monoclonal antibody or 28 days have elapsed since last dose of the monoclonal antibody with complete resolution of symptoms related to treatment. Radiotherapy: At least 2 weeks must have elapsed since any irradiation; at least 6 weeks must have elapsed since craniospinal RT, 131I-mIBG therapy or substantial bone marrow irradiation (e.g., >50% pelvis irradiation). Female participant who is post-menarchal must have a negative urine or serum pregnancy test and must be willing to have additional serum and urine pregnancy tests during the study. Female or male participant of reproductive potential must agree to use effective contraceptive methods at screening and throughout duration of study treatment.  Exclusion Criteria  Pregnant or breastfeeding  Pregnant or breast-feeding women will not be entered on this study. Pregnancy tests must be obtained in girls who are post-menarchal. Males or females of reproductive potential may not participate unless they have agreed to use two methods of birth control: a medically accepted barrier of contraceptive method (e.g., male or female condom) and a second method of birth control during protocol therapy. Two highly effective methods of contraception are required for female patients during treatment and for at least 7 months after completing therapy. Male patients with female partners of reproductive potential and/or pregnant partners are advised to use two highly effective methods of contraception during treatment and for at least 4 months after the final dose. Male and female participants must agree not to donate sperm or eggs, respectively, after the first dose of study drug through 105 days and 45 days after the last dose of study drug. Females considered not of childbearing potential include those who are surgically sterile (bilateral salpingectomy, bilateral oophorectomy, or hysterectomy).