DNA topoisomerases are complex and unique enzymes which alter the topological state of DNA without changing its chemical structure. Between the type I and II enzymes, topoisomerases carry out a multitude of reactions, including DNA binding, site specific DNA cleavage/religation, relaxation, catenation/decatenation, and knotting/unknotting of nucleic acid substrates, DNA strand transfer, and ATP hydrolysis. In vivo, topoisomerases are involved in many aspects of nucleic acid metabolism and play critical roles in maintaining chromosome and nuclear structure. Finally, these enzymes are of clinical relevance, as they appear to be the primary cellular targets for many varied classes of antineoplastic agents. Considering the importance of the topoisomerases, it is distressing that we know so little about their enzymatic mechanisms. Many major questions remain. Just a few include, "How do topoisomerases recognize their nucleic acid interaction sites?"; "What amino acid residues comprise the enzymes' active sites?"; "What are the conformational changes that accompany DNA strand passage?"; "How does phosphorylation stimulate enzyme activity?"; "How does topoisomerase function when it is part of an immobilized structure such as the nuclear matrix or the mitotic chromosome scaffold?"; and "How do antineoplastic agents interact with their topoisomerase targets and stabilize covalent enzyme.DNA cleavage products?" Clearly, before the physiological functions of the topoisomerases can be fully described, these and similar issues will have to be addressed. Hopefully, the next several years will produce answers for at least some of these important questions.