In this talk I will report our recent works on convex hypersurfaces in hyperbolic space. To study hypersurfaces in hyperbolic space analytically, one needs to find ways to parametrize it, preferably globally. We consider two parametrizations: vertical graph and hyperbolic Gauss map. To get a global parametrization, one needs understand the interrelation of convexity and embeddedness. It is also important to understand the asymptotic of the geometry at ends. In this talk I will report some of our recent works on global and asymptotic properties of hypersurfaces with nonnegative sectional curvature or Ricci curvature in hyperbolic space, where our use of n-Laplace equations seems to be new.