We report the growth of GaAs/AlGaAs core-shell nanowires (NWs) on GaAs(111)B substrates by Au-assisted molecular beam epitaxy. Electron microscopy shows the formation of a wurtzite AlGaAs shell structure both in the radial and the axial directions outside a wurtzite GaAs core. With higher Al content, a lower axial and a higher radial growth rate of the AlGaAs shell were observed. Room temperature and low temperature (4.4 K) micro-photoluminescence measurements show a much higher radiative efficiency from the GaAs core after the NW is overgrown with a radial AlGaAs shell.