We employ gravitational-wave radiometry to map the stochastic gravitational wave background expected from a variety of contributing mechanisms and test the assumption of isotropy using data from the Advanced Laser Interferometer Gravitational Wave Observatory's (aLIGO) first observing run. We also search for persistent gravitational waves from point sources with only minimal assumptions over the 20-1726 Hz frequency band. Finding no evidence of gravitational waves from either point sources or a stochastic background, we set limits at 90% confidence. For broadband point sources, we report upper limits on the gravitational wave energy flux per unit frequency in the range Fα,Θ(f)<(0.1-56)×10-8 erg cm-2 s-1 Hz-1(f/25 Hz)α-1 depending on the sky location Θ and the spectral power index α. For extended sources, we report upper limits on the fractional gravitational wave energy density required to close the Universe of Ω(f,Θ)<(0.39-7.6)×10-8 sr-1(f/25 Hz)α depending on Θ and α. Directed searches for narrowband gravitational waves from astrophysically interesting objects (Scorpius X-1, Supernova 1987 A, and the Galactic Center) yield median frequency-dependent limits on strain amplitude of h0<(6.7,5.5, and 7.0)×10-25, respectively, at the most sensitive detector frequencies between 130-175 Hz. This represents a mean improvement of a factor of 2 across the band compared to previous searches of this kind for these sky locations, considering the different quantities of strain constrained in each case.