DNA repair enzyme that can remove a variety of covalent adducts from DNA through hydrolysis of a 3'-phosphodiester bond, giving rise to DNA with a free 3' phosphate. Catalyzes the hydrolysis of dead- end complexes between DNA and the topoisomerase I active site tyrosine residue. Hydrolyzes 3'-phosphoglycolates on protruding 3' ends on DNA double-strand breaks due to DNA damage by radiation and free radicals. Acts on blunt-ended double-strand DNA breaks and on single-stranded DNA. Has low 3'exonuclease activity and can remove a single nucleoside from the 3'end of DNA and RNA molecules with 3'hydroxyl groups. Has no exonuclease activity towards DNA or RNA with a 3'phosphate. {ECO:0000269|PubMed:12023295, ECO:0000269|PubMed:15111055, ECO:0000269|PubMed:15811850, ECO:0000269|PubMed:16141202, ECO:0000269|PubMed:22822062}. This is the function of TDP1 (tyrosyl-DNA phosphodiesterase 1, ENSG00000042088).