  Black hole-neutron star (BHNS) binary mergers are candidate engines for
generating both short-hard gamma-ray bursts (SGRBs) and detectable
gravitational waves. Using our most recent conformal thin-sandwich BHNS initial
data and our fully general relativistic hydrodynamics code, which is now
AMR-capable, we are able to efficiently and accurately simulate these binaries
from large separations through inspiral, merger, and ringdown. We evolve the
metric using the BSSN formulation with the standard moving puncture gauge
conditions and handle the hydrodynamics with a high-resolution shock-capturing
scheme. We explore the effects of BH spin (aligned and anti-aligned with the
orbital angular momentum) by evolving three sets of initial data with BH:NS
mass ratio q=3: the data sets are nearly identical, except the BH spin is
varied between a/M = -0.5 (anti-aligned), 0.0, and 0.75. The number of orbits
before merger increases with a/M, as expected. We also study the nonspinning BH
case in more detail, varying q between 1, 3, and 5. We calculate gravitational
waveforms for the cases we simulate and compare them to binary black-hole
waveforms. Only a small disk (< 0.01 M_sun) forms for the anti-aligned spin
case (a/M = -0.5) and for the most extreme mass ratio case (q=5). By contrast,
a massive (M_disk is about 0.2 M_sun), hot disk forms in the rapidly spinning
(a/M = 0.75) aligned BH case. Such a disk could drive a SGRB,possibly by, e.g.,
producing a copious flux of neutrino-antineutino pairs.
