In this talk I will introduce a global correspondence between properly immersed horospherically convex hyper surfaces in hyperbolic space and complete conformal metrics on subdomains in the boundary at infinity of hyperbolic space. I will discuss when a horospherically convex hypersurface is proper,  when its hyperbolic Gauss map is injective, and when it is embedded. These are expected to be useful to the understandings of both elliptic problems of Weingarten hypersurfaces in hyperbolic space and elliptic problems of complete conformal metrics on subdomains in sphere.