Gravitational waves from the coalescence of two neutron stars were detected for the first time in August. The detection of both gravitational and electromagnetic radiation from a single astrophysical source showcased the uniqueness and importance of these systems as probes of astrophysics, nuclear physics, and fundamental theory. In this talk I will describe the properties of the gravitational wave signal emitted when two neutron stars merge and what we can learn from each stage of the coalescence. I will argue that in order to efficiently characterize these complicated systems we need to combine knowledge of astrophysics, nuclear physics, general relativity, and data analysis; I will discuss three relevant examples. Finally, I will discuss my future plans for improvements in the analysis and understanding of neutron star coalescences.