  We present a new catalog of TeV gamma-ray sources using 1523 days of data
from the High Altitude Water Cherenkov (HAWC) observatory. The catalog
represents the most sensitive survey of the Northern gamma-ray sky at energies
above several TeV, with three times the exposure compared to the previous HAWC
catalog, 2HWC. We report 65 sources detected at $\geq$ 5 sigma significance,
along with the positions and spectral fits for each source. The catalog
contains eight sources that have no counterpart in the 2HWC catalog, but are
within $1^\circ$ of previously detected TeV emitters, and twenty sources that
are more than $1^\circ$ away from any previously detected TeV source. Of these
twenty new sources, fourteen have a potential counterpart in the fourth
\textit{Fermi} Large Area Telescope catalog of gamma-ray sources. We also
explore potential associations of 3HWC sources with pulsars in the ATNF pulsar
catalog and supernova remnants in the Galactic supernova remnant catalog.
