Refreshments served 30 minutes prior to start of talk in front of the JILA Auditorium. Classically, in the range of a few tens of percent of Eddington, black hole accretion disks are predicted to be thermally (and viscously) unstable. This stems from the fact that the inner regions of disks at these luminosities are dependent upon radiation pressure to provide the vertical support against gravity. However, in radiation-pressure-dominated accretion disks, heating and cooling depend on different powers of the temperature, leading to the thermal instability. In recent global, general relativistic numerical simulations, we have verified this instability by seeing disks collapse vertically on the local thermal timescale. Of course, these results are not at all surprising, as they confirm the 40 year old prediction, yet they appear to be in conflict with 20+ years of observations of accreting black holes, which show disks with luminosities in this range to be exceedingly stable. I will discuss possible solutions to this dilemma and our attempts to test them numerically.