TDP1 and TDP2 were discovered and named based on the fact they process 3'- and 5'-DNA ends by excising irreversible protein tyrosyl-DNA complexes involving topoisomerases I and II, respectively. Yet, both enzymes have an extended spectrum of activities. TDP1 not only excises trapped topoisomerases I (Top1 in the nucleus and Top1mt in mitochondria), but also repairs oxidative damage-induced 3'-phosphoglycolates and alkylation damage-induced DNA breaks, and excises chain terminating anticancer and antiviral nucleosides in the nucleus and mitochondria. The repair function of TDP2 is devoted to the excision of topoisomerase II- and potentially topoisomerases III-DNA adducts. TDP2 is also essential for the life cycle of picornaviruses (important human and bovine pathogens) as it unlinks VPg proteins from the 5'-end of the viral RNA genome. Moreover, TDP2 has been involved in signal transduction (under the former names of TTRAP or EAPII). The DNA repair partners of TDP1 include PARP1, XRCC1, ligase III and PNKP from the base excision repair (BER) pathway. By contrast, TDP2 repair functions are coordinated with Ku and ligase IV in the non-homologous end joining pathway (NHEJ). This article summarizes and compares the biochemistry, functions, and post-translational regulation of TDP1 and TDP2, as well as the relevance of TDP1 and TDP2 as determinants of response to anticancer agents. We discuss the rationale for developing TDP inhibitors for combinations with topoisomerase inhibitors (topotecan, irinotecan, doxorubicin, etoposide, mitoxantrone) and DNA damaging agents (temozolomide, bleomycin, cytarabine, and ionizing radiation), and as novel antiviral agents.