The extraordinary properties and the novel applications of black phosphorene induce the research interest in the monolayer group-IV monochalcogenides. Here using first-principles calculations, we systematically investigate the electronic, transport, and optical properties of monolayer alpha- and beta-GeSe, revealing a direct band gap of 1.61 eV for monolayer alpha-GeSe and an indirect band gap of 2.47 eV for monolayer beta-GeSe. For monolayer beta-GeSe, the electronic/hole transport is anisotropic, with an extremely high electron mobility of 2.93 x 10(4) cm(2)/Vs along the armchair direction, comparable to that of black phosphorene. Furthermore, for beta-GeSe, robust band gaps nearly independent of the applied tensile strain along the armchair direction are observed. Both monolayer aand beta-GeSe exhibit anisotropic optical absorption in the visible spectrum.