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 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 Fα,Θ < (0.013–7.6) ×10−8 erg cm−2 s−1 Hz−1, and on the (normalized) gravitational-wave energy density spectrum from extended sources, ranging from Ωα,Θ < (0.57–9.3) ×10−9 sr−1, depending on direction (Θ) and spectral index (α). 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 h0 < (1.7–2.1) ×10−25, a factor of ≥ 2.0 improvement compared to previous stochastic radiometer searches.