  We present the general theory of curves in conformal geometry using tractor
calculus. This primarily involves a tractorial determination of distinguished
parametrizations and relative and absolute conformal invariants of generic
curves. The absolute conformal invariants are defined via a tractor analogue of
the classical Frenet frame construction and then expressed in terms of relative
ones. This approach applies likewise to conformal structures of any signature;
in the case of indefinite signature we focus especially on the null curves. It
also provides a conceptual tool for handling distinguished families of curves
(conformal circles and conformal null helices) and conserved quantities along
them.
