Core-collapse supernovae (CCSNe) mark the violent deaths of massive stars and are among the most energetic explosions in the Universe. These catastrophic events are responsible for the formation of the neutron stars and black holes observed through gravitational waves. CCSNe also directly emit gravitational waves, and are key targets for gravitational-wave detection, offering a unique window into the central mechanism that powers these explosions. This talk will discuss the latest developments in CCSN numerical simulations, including the impact on the birth properties of black holes and neutron stars observed through gravitational waves, the latest CCSN gravitational waveform models, and the potential for current and next-generation gravitational-wave detectors to extract astrophysical and nuclear physics insights from these events.