  The equivalence problem of curves with values in a Riemannian manifold, is
solved. The domain of validity of Frenet's theorem is shown to be the spaces of
constant curvature. For a general Riemannian manifold new invariants must thus
be added. There are two important generic classes of curves; namely, Frenet
curves and a new class, called curves "in normal position". They coincide in
dimensions $\leq 4$ only. A sharp bound for asymptotic stability of
differential invariants is obtained, the complete systems of invariants are
characterized, and a procedure of generation is presented. Different classes of
examples (specially in low dimensions) are analyzed in detail.
