One of the most interesting and striking concepts in Differential Geometry is that of the Gauss map. In the case of surfaces, this map projects surface normals to a unit sphere. This strategy is especially useful when analyzing the shape structure of a smooth surface. This paper describes a new Mathematica package, GaussMap, for computing and displaying the tangent and normal vector fields and the Gauss map of surfaces described symbolically in either implicit or parametric form. The performance of the package is discussed by means of several illustrative and interesting examples. The package presented here can be applied for visualizing and studying the geometry of a surface under analysis, thus providing the users with an excellent computer tool for teaching and visualization purposes.