The classical Gauss map \(\gamma\) : \(X^ n\to G(N,n)\) associates to a point x of a nonsingular projective algebraic variety \(X^ n\subset {\mathbb{P}}^ N\) the point in the Grassmann variety G(N,n) of n-dimensional linear subspaces in \({\mathbb{P}}^ N\) corresponding to the embedded tangent space \(T_{X,x}\) to X at x. Thus the fiber of \(\gamma\) over a point \(L\in G(N,n)\) is the set of points (with multiplicities) at which the embedded tangent space to X coincides with L. Similarly, for each \(n\leq m\leq N-1\) we consider the higher Gauss map \(\gamma_ m\) whose fiber over a point \(L\in G(N,m)\) coincides with the set of points \(x\in X\) such that \(T_{X,x}\subset L^ m\) (i.e. L is tangent to X at x). We study the structure of the maps \(\gamma_ m\) with special reference to the cases \(m=n\) and \(m=N-1\) and consider applications to tangencies, projections, varieties of small codimension etc.