  The High Altitude Water Cherenkov (HAWC) Gamma-ray Observatory is an
extensive air shower detector operating in central Mexico, which has recently
completed its first two years of full operations. If for a burst like GRB
130427A at a redshift of 0.34 and a high-energy component following a power law
with index -1.66, the high-energy component is extended to higher energies with
no cut-off other than from extragalactic background light attenuation, HAWC
would observe gamma rays with a peak energy of $\sim$300 GeV. This paper
reports the results of HAWC observations of 64 gamma-ray bursts (GRBs) detected
by $\mathit{Swift}$ and $\mathit{Fermi}$, including three GRBs that were also
detected by the Large Area Telescope ($\mathit{Fermi}$-LAT). An ON/OFF analysis
method is employed, searching on the time scale given by the observed light
curve at keV-MeV energies and also on extended time scales. For all GRBs and
time scales, no statistically significant excess of counts is found and upper
limits on the number of gamma rays and the gamma-ray flux are calculated. GRB
170206A, the third brightest short GRB detected by the Gamma-ray Burst Monitor
on board the $\mathit{Fermi}$ satellite ($\mathit{Fermi}$-GBM) and also
detected by the LAT, occurred very close to zenith. The LAT measurements can
neither exclude the presence of a synchrotron self-Compton (SSC) component nor
constrain its spectrum. Instead, the HAWC upper limits constrain the expected
cut-off in an additional high-energy component to be less than $100~\rm{GeV}$
for reasonable assumptions about the energetics and redshift of the burst.
