  In this paper we prove quantitative results about geodesic approximations to
submanifolds in negatively curved spaces. Among the main tools is a new and
general Jarn\'{i}k-Besicovitch type theorem in Diophantine approximation. The
framework we develop is flexible enough to treat manifolds of variable negative
curvature, a variety of geometric targets, and logarithm laws as well as
spiraling phenomena in both measure and dimension aspect. Several of the
results are new also for manifolds of constant negative sectional curvature. We
further establish a large intersection property of Falconer in this context.
