For a simple Lie group G and a discrete subgroup Gamma of G, we will discuss exponential decay of the matrix coefficients of G associated to functions on the quotient space Gamma\G. When G is of rank one, and Gamma is a lattice, this yields exponential mixing of the geodesic flow for the associated Riemannian manifold. We will also discuss a uniform exponential decay for congruence subgroups of Gamma when Gamma is contained in an arithmetic subgroup of G.