  We present the results of targeted searches for gravitational-wave transients
associated with gamma-ray bursts during the second observing run of Advanced
LIGO and Advanced Virgo, which took place from 2016 November to 2017 August. We
have analyzed 98 gamma-ray bursts using an unmodeled search method that
searches for generic transient gravitational waves and 42 with a modeled search
method that targets compact-binary mergers as progenitors of short gamma-ray
bursts. Both methods clearly detect the previously reported binary merger
signal GW170817, with p-values of $<9.38 \times 10^{-6}$ (modeled) and $3.1
\times 10^{-4}$ (unmodeled). We do not find any significant evidence for
gravitational-wave signals associated with the other gamma-ray bursts analyzed,
and therefore we report lower bounds on the distance to each of these, assuming
various source types and signal morphologies. Using our final modeled search
results, short gamma-ray burst observations, and assuming binary neutron star
progenitors, we place bounds on the rate of short gamma-ray bursts as a
function of redshift for $z \leq 1$. We estimate 0.07-1.80 joint detections
with Fermi-GBM per year for the 2019-20 LIGO-Virgo observing run and 0.15-3.90
per year when current gravitational-wave detectors are operating at their
design sensitivities.
