Since the report of “DNA untwisting” activity in 1972, ∼50 years of research has revealed seven topoisomerases in humans (TOP1, TOP1mt, TOP2α, TOP2β, TOP3α, TOP3β and Spo11). These conserved regulators of DNA topology catalyze controlled breakage to the DNA backbone to relieve the torsional stress that accumulates during essential DNA transactions including DNA replication, transcription, and DNA repair. Each topoisomerase-catalyzed reaction involves the formation of a topoisomerase cleavage complex (TOPcc), a covalent protein-DNA reaction intermediate formed between the DNA phosphodiester backbone and a topoisomerase catalytic tyrosine residue. A variety of perturbations to topoisomerase reaction cycles can trigger failure of the enzyme to re-ligate the broken DNA strand(s), thereby generating topoisomerase DNA-protein crosslinks (TOP-DPC). TOP-DPCs pose unique threats to genomic integrity. These complex lesions are comprised of structurally diverse protein components covalently linked to genomic DNA, which are bulky DNA adducts that can directly impact progression of the transcription and DNA replication apparatus. A variety of genome maintenance pathways have evolved to recognize and resolve TOP-DPCs. Eukaryotic cells harbor tyrosyl DNA phosphodiesterases (TDPs) that directly reverse 3′-phosphotyrosyl (TDP1) and 5′-phoshotyrosyl (TDP2) protein-DNA linkages. The broad specificity Mre11-Rad50-Nbs1 and APE2 nucleases are also critical for mitigating topoisomerase-generated DNA damage. These DNA-protein crosslink metabolizing enzymes are further enabled by proteolytic degradation, with the proteasome, Spartan, GCNA, Ddi2, and FAM111A proteases implicated thus far. Strategies to target, unfold, and degrade the protein component of TOP-DPCs have evolved as well. Here we survey mechanisms for addressing Topoisomerase 1 (TOP1) and Topoisomerase 2 (TOP2) DPCs, highlighting systems for which molecular structure information has illuminated function of these critical DNA damage response pathways.