  We report results from searches for anisotropic stochastic gravitational-wave
backgrounds using data from the first three observing runs of the Advanced LIGO
and Advanced Virgo detectors. For the first time, we include Virgo data in our
analysis and run our search with a new efficient pipeline called {\tt PyStoch}
on data folded over one sidereal day. We use gravitational-wave radiometry
(broadband and narrow band) to produce sky maps of stochastic
gravitational-wave backgrounds and to search for gravitational waves from point
sources. A spherical harmonic decomposition method is employed to look for
gravitational-wave emission from spatially-extended sources. Neither technique
found evidence of gravitational-wave signals. Hence we derive 95\%
confidence-level upper limit sky maps on the gravitational-wave energy flux
from broadband point sources, ranging from $F_{\alpha, \Theta} < {\rm (0.013 -
7.6)} \times 10^{-8} {\rm erg \, cm^{-2} \, s^{-1} \, Hz^{-1}},$ and on the
(normalized) gravitational-wave energy density spectrum from extended sources,
ranging from $\Omega_{\alpha, \Theta} < {\rm (0.57 - 9.3)} \times 10^{-9} \,
{\rm sr^{-1}}$, depending on direction ($\Theta$) and spectral index
($\alpha$). These limits improve upon previous limits by factors of $2.9 -
3.5$. We also set 95\% confidence level upper limits on the frequency-dependent
strain amplitudes of quasimonochromatic gravitational waves coming from three
interesting targets, Scorpius X-1, SN 1987A and the Galactic Center, with best
upper limits range from $h_0 < {\rm (1.7-2.1)} \times 10^{-25},$ a factor of
$\geq 2.0$ improvement compared to previous stochastic radiometer searches.
