Fully general relativistic simulations of black hole–neutron star (BHNS) binary inspiral and merger indicate that the NS simply plunges into the BH over much of the likely binary parameter space, without leaving a remnant disk. Plunging mergers are unlikely to generate useful information about the NS equation of state (EOS) from the gravitational waves (GWs) alone. However, when the initial BH possesses a moderate to high spin, aligned with the orbital angular momentum, or when the NS possesses a low compaction, the NS may tidally disrupt outside the BH’s innermost stable circular orbit radius, generating a massive accretion disk and a long tidal tail. When observed by Advanced LIGO/VIRGO, the GWs from this scenario may in fact constrain the NS EOS. After disruption, some of the neutron-rich tidal tail may be unbound, favoring formation of r-process elements. The subsequent decay may be observable in the electromagnetic (EM) spectrum. Meanwhile, the remnant BH accretion disk may provide the energy reservoir for a short gamma-ray burst. Finally, even in cases for which no EOS informationmay be gleaned, the BHNS binary is expected to act as a unipolar inductor during inspiral, potentially releasing an EM signature that distinguishes certain binary parameters. Taken together, these phenomena make BHNS binary mergers rich and exciting systems for theoretical modeling, and we outline the latest results from the most advanced, fully general relativistic simulations.