We perform an unmodeled search for persistent, directional gravitational wave (GW) sources using data from the first and second observing runs of Advanced LIGO. We do not find evidence for any GW signals. We place limits on the broadband GW flux emitted at 25 Hz from point sources with a power law spectrum at Fα,Θ<(0.05-25)×10-8 erg cm-2 s-1 Hz-1 and the (normalized) energy density spectrum in GWs at 25 Hz from extended sources at ωα(Θ)<(0.19-2.89)×10-8 sr-1 where α is the spectral index of the energy density spectrum. These represent improvements of 2.5-3× over previous limits. We also consider point sources emitting GWs at a single frequency, targeting the directions of Sco X-1, SN 1987A, and the Galactic center. The best upper limits on the strain amplitude of a potential source in these three directions range from h0<(3.6-4.7)×10-25, 1.5× better than previous limits set with the same analysis method. We also report on a marginally significant outlier at 36.06 Hz. This outlier is not consistent with a persistent gravitational-wave source as its significance diminishes when combining all of the available data.