We characterize strongly Morse quasi-geodesics in Outer space as quasi-geodesics which project to quasi-geodesics in the free factor graph. We define convex cocompact subgroups of Out (F_n) as subgroups such that an orbit map in the free factor graph is a quasi-isometric embedding, and we characterize such groups via their action on Outer space in a way which resembles the characterization of convex cocompact subgroups of mapping class groups.