This phase 2 trial explores the activity of single agent talazoparib in BRCA1/2 wild-type BC patients using an optimal Simon two-stage design. Eligible subjects will be assigned to one of two parallel cohorts: 1) Cohort A: Subjects (n=29) with advanced TNBC with underlying HR defects as assessed by the HRD assay and, 2) Cohort B: Subjects (n=29) with advanced HER2-negative BC with a somatic or germline mutation in a non-BRCA1/2 HR pathway gene. Gene mutations of interest are: PTEN, PALB2, CHEK2, ATM, NBN, BARD1, BRIP1, RAD50, RAD51C, RAD51D, MRE11, ATR, Fanconi anemia complementation group of genes (FANCA, FANCC, FANCD2, FANCE, FANCF, FANCG, FANCL).