The kinetic mechanisms by which DNA polymerases catalyze DNA replication and repair have long been areas of active research. Recently discovered Y-family DNA polymerases catalyze the bypass of damaged DNA bases that would otherwise block replicative DNA polymerases and stall replication forks. Unlike DNA polymerases from the five other families, the Y-family DNA polymerases have flexible, solvent-accessible active sites that are able to tolerate various types of damaged template bases and allow for efficient lesion bypass. Their promiscuous active sites, however, also lead to fidelities that are much lower than those observed for other DNA polymerases and give rise to interesting mechanistic properties. Additionally, the Y-family DNA polymerases have several other unique structural features and undergo a set of conformational changes during substrate binding and catalysis different from those observed for replicative DNA polymerases. In recent years, pre-steady-state kinetic methods have been extensively employed to reveal a wealth of information about the catalytic properties of these fascinating noncanonical DNA polymerases. Here, we review many of the recent findings on the kinetic mechanisms of DNA polymerization with undamaged and damaged DNA substrates by the Y-family DNA polymerases, and the conformational dynamics employed by these error-prone enzymes during catalysis.