  We present a search for gravitational waves from the coalescence of binaries
which contain at least one subsolar mass component using data from the LIGO and
Virgo observatories through the completion of their third observing run. The
observation of a merger with a component below $1\,M_{\odot}$ would be a clear
sign of either new physics or the existence of a primordial black hole
population; these black holes could also contribute to the dark matter
distribution. Our search targets binaries where the primary has mass $M_1$
between 0.1-100$\,M_{\odot}$ and the secondary has mass $M_2$ from
0.1-1$\,M_{\odot}$ for $M_1 < 20\,M_\odot$ and 0.01-1$\,M_{\odot}$ for $M_1 \ge
20\,M_\odot$. Sources with $M_1 < 7\,M_\odot, M_2 > 0.5\,M_\odot$ are also
allowed to have orbital eccentricity up to $e_{10} \sim 0.3$. This search
region covers from comparable to extreme mass ratio sources up to $10^4:1$. We
find no statistically convincing candidates and so place new upper limits on
the rate of mergers; our analysis sets the first limits for most subsolar
sources with $7\,M_{\odot}< M_1 < 20\,M_{\odot}$ and tightens limits by $\sim
8\times$ $(1.6\times)$ where $M_1 > 20\,M_{\odot}$ ($M_1 < 7\,M_{\odot}$).
Using these limits, we constrain the dark matter fraction to below 0.3
(0.7)$\%$ for 1 (0.5)$\,M_{\odot}$ black holes assuming a monochromatic mass
function. Due to the high merger rate of primordial black holes beyond the
individual source horizon distance, we also use the lack of an observed
stochastic background as a complementary probe to limit the dark matter
fraction. We find that although the limits are in general weaker than those
from the direct search they become comparable at $0.1 \,M_{\odot}$.
