The large GTPase dynamin is the first protein shown to catalyze membrane fission. Dynamin and its related proteins are essential to many cell functions, from endocytosis to organelle division and fusion, and it plays a critical role in many physiological functions such as synaptic transmission and muscle contraction. Research of the past three decades has focused on understanding how dynamin works. In this review, we present the basis for an emerging consensus on how dynamin functions. Three properties of dynamin are strongly supported by experimental data: first, dynamin oligomerizes into a helical polymer; second, dynamin oligomer constricts in the presence of GTP; and third, dynamin catalyzes membrane fission upon GTP hydrolysis. We present the two current models for fission, essentially diverging in how GTP energy is spent. We further discuss how future research might solve the remaining open questions presently under discussion.