In this paper we adopt the Hyperboloid in Minkowski space as the model of Hyperbolic space. We define the hyperbolic Gauss map and the hyperbolic Gauss indicatrix of a hypersurface in Hyperbolic space. The hyperbolic Gauss map has been introduced by Epstein[7]in the Poincar@ball model which is very useful for the study of constant mean curvature surfaces. However, it iS very hard to proceed the calculation because it has an intrinsic form. Here, we give an extrinsic definition and we study singularities of these. In the study of singularities of the hyperbolic Gauss map(indicatrix), we understand that the hyperbolic Gauss indicatrix is much easier to proceed the calculation. We introduce, the notion of hyperbolic Gauss-Kronecker curvature whose zero sets correspond to the singular set of the hyperbolic GaUSS map(indicatrix). We also develop a 10cal differential geometry of hypersurfaces concerning on contact with hyperhorospheres.