Stars orbiting sufficiently close to massive black holes inevitably inspiral towards the black hole due to gravitational wave radiation. The fate of such stars is subtle and depends on an interplay between tidal heating of the star, mass transfer from the star to the black hole, and collisions between the star and any ambient accretion disk. In this talk, I will describe the possible fates of stars on such orbits and their connection to enigmatic classes of recently discovered recurring transients in galactic nuclei, including partial tidal disruption events and quasi-periodic eruptions. I will also discuss the connection between inspiraling stars and other stellar dynamical processes in galactic nuclei, e.g., the orbiting stars observed in the Galactic Center, extreme mass ratio inspirals of compact objects observable by LISA, hyper-velocity stars, and tidal disruption events.