DNA topoisomerases are a diverse set of essential enzymes responsible for maintaining chromosomes in an appropriate topological state. Although they vary considerably in structure and mechanism, the partnership between topoisomerases and DNA has engendered commonalities in how these enzymes engage nucleic acid substrates and control DNA strand manipulations. All topoisomerases can harness the free energy stored in supercoiled DNA to drive their reactions; some further use the energy of ATP to alter the topology of DNA away from an enzyme-free equilibrium ground state. In the cell, topoisomerases regulate DNA supercoiling and unlink tangled nucleic acid strands to actively maintain chromosomes in a topological state commensurate with particular replicative and transcriptional needs. To carry out these reactions, topoisomerases rely on dynamic macromolecular contacts that alternate between associated and dissociated states throughout the catalytic cycle. In this review, we describe how structural and biochemical studies have furthered our understanding of DNA topoisomerases, with an emphasis on how these complex molecular machines use interfacial interactions to harness and constrain the energy required to manage DNA topology.