  While optical surveys regularly discover slow transients like supernovae on
their own, the most common way to discover extragalactic fast transients,
fading away in a few nights, is via follow-up observations of gamma-ray burst
and gravitational-wave triggers. However, wide-field surveys have the potential
to also identify rapidly fading transients independently of such external
triggers. The volumetric survey speed of the Zwicky Transient Facility (ZTF)
makes it sensitive to faint and fast-fading objects as kilonovae, the optical
counterparts to binary neutron stars and neutron star-black hole mergers, out
to almost 200Mpc. We introduce an open-source software infrastructure, the ZTF
REaltime Search and Triggering, ZTFReST, designed to identify kilonovae and
fast optical transients in ZTF data. Using the ZTF alert stream combined with
forced photometry, we have implemented automated candidate ranking based on
their photometric evolution and fitting to kilonova models. Automated
triggering of follow-up systems, such as Las Cumbres Observatory, has also been
implemented. In 13 months of science validation, we found several extragalactic
fast transients independent of any external trigger (though some counterparts
were identified later), including at least one supernova with post-shock
cooling emission, two known afterglows with an associated gamma-ray burst, two
known afterglows without any known gamma-ray counterpart, and three new
fast-declining sources (ZTF20abtxwfx, ZTF20acozryr, and ZTF21aagwbjr) that are
likely associated with GRB200817A, GRB201103B, and GRB210204A. However, we have
not found any objects which appear to be kilonovae; therefore, we constrain the
rate of GW170817-like kilonovae to $R < 900$Gpc$^{-3}$yr$^{-1}$. A framework
such as ZTFReST could become a prime tool for kilonova and fast transient
discovery with the Vera C. Rubin Observatory.
