In recent years, dozens of powerful, multiwavelength transients have been seen from the photometric centers of inactive galactic nuclei. The light curves of these explosions rise and decay over timescales of weeks to months, their generally thermal spectra peak in the UV or soft X-ray, and their bolometric energy output is greater than the vast majority of supernovae. I will review the observational evidence that these flares are tidal disruption events (TDEs), the death throes of stars that have been torn apart by supermassive black holes (SMBHs). I will also present my own, primarily theoretical, work to understand the surprisingly complex physics underlying TDEs. This is a field with important open questions: what are the stellar dynamical processes that feed stars to SMBHs? How do eccentric debris streams from the disrupted star circularize into an accretion disk? What are the emission mechanisms that power the flares we observe? I will propose answers to each of these puzzles, and highlight upcoming observational tests that may answer them decisively. Although the current observational sample of TDEs is modest, upcoming time domain surveys such as ZTF, eROSITA, and LSST will increase our sample size by orders of magnitude. Resolving the theoretical questions I will discuss in this colloquium is the critical step toward using these flares as unparalleled probes of SMBH demography.