The authors study the differential geometry of hypersurfaces in hyperbolic space. They reformulate the questions and adapt the methods coming from the classical theory of Gauss map of a surface in Euclidean 3-space. They study singularities of hyperbolic Gauss map by adapting the hyperboloid in Minkowski space as the model of hyperbolic space, and using the local parametrization of the hypersurface. Redefining of the notion of the light-cone normal and the hyperbolic Gauss indicatrix of a hypersurface in hyperbolic space they study two hyperbolic invariants; the hyperbolic Gauss-Kronecker curvature and the hyperbolic mean curvature. The theory of Legendrian singularities is applied to study hyperbolic Gauss indicatrices in connection to the classical property of the Gauss-map to be a Lagrangian map.