  All parabolic geometries, i.e. Cartan geometries with homogeneous model a
real generalized flag manifold, admit highly interesting classes of
distinguished curves. The geodesics of a projective class of connections on a
manifold, conformal circles on conformal Riemannian manifolds, and Chern--Moser
chains on CR--manifolds of hypersurface type are typical examples. We show that
such distinguished curves are always determined by a finite jet in one point,
and study the properties of such jets. We also discuss the question when
distinguished curves agree up to reparametrization and discuss the
distinguished parametrizations in this case. We give a complete description of
all distinguished curves for some examples of parabolic geometries.
