  Causal rigid particles whose action includes an {\it arbitrary} dependence on
the world-line extrinsic curvature are considered. General classes of solutions
are constructed, including {\it causal tachyonic} ones. The Hamiltonian
formulation is developed in detail except for one degenerate situation for
which only partial results are given and requiring a separate analysis.
However, for otherwise generic rigid particles, the precise specification of
Hamiltonian gauge symmetries is obtained with in particular the identification
of the Teichm$\ddot{\rm u}$ller and modular spaces for these systems. Finally,
canonical quantisation of the generic case is performed paying special
attention to the phase space restriction due to causal propagation. A mixed
Lorentz-gravitational anomaly is found in the commutator of Lorentz boosts with
world-line reparametrisations. The subspace of gauge invariant physical states
is therefore not invariant under Lorentz transformations. Consequences for
rigid strings and membranes are also discussed.
