Astronomers have discovered many excellent black hole candidates in X-ray binaries and galactic nuclei. These black holes are identified by the fact that they are ultra-compact and too massive to be neutron stars. Can we do better? Can we demonstrate that black hole candidates really are black holes, with event horizons?

The talk will describe recent work in this area, highlighting a class of accretion models called advection-dominated accretion flows. In these flows, the black hole nature of the accreting star, specifically its event horizon, plays a critical role, and so any agreement between the model predictions and observations may be viewed as confirmation of the reality of black hole horizons. The model has been compared to spectral data on several sources. The current status of the field will be reviewed.