Abstract: In this talk we shall derive several results that describe the rate at which a geodesic makes excusions into and out of a cusp on a finite area hyperbolic surface and relate them to approximation with respect to the orbit of infinity for an associated Fuchsian group. This provides proofs of several important theorems from metric diophantine approximation in the context of Fuchsian groups. It also illuminates the classical theory and produces new estimates on the rate of growth of the partial quotients in the continued fraction expansion of a generic real number.