  We study spaces with a cuspidal (or horn-like) singularity embedded in a
smooth Riemannian manifold and analyze the geodesics in these spaces which
start at the singularity. This provides a basis for understanding the intrinsic
geometry of such spaces near the singularity. We show that these geodesics
combine to naturally define an exponential map based at the singularity, but
that the behavior of this map can deviate strongly from the behavior of the
exponential map based at a smooth point or at a conical singularity: While it
is always surjective near the singularity, it may be discontinuous and
non-injective on any neighborhood of the singularity. The precise behavior of
the exponential map is determined by a function on the link of the singularity
which is an invariant -- essentially the only boundary invariant -- of the
induced metric. Our methods are based on the Hamiltonian system of geodesic
differential equations and on techniques of singular analysis. The results are
proved in the more general natural setting of manifolds with boundary carrying
a so-called cuspidal metric.
  Revision: restructured many parts of the paper completely; stated main
theorems for cusps of any order; replaced 'cusp' by 'cuspidal', also in title,
added references
