  We define a L\'evy process on a smooth manifold $M$ with a connection as a
projection of a solution of a Marcus stochastic differential equation on a
holonomy bundle of $M$, driven by a holonomy-invariant L\'evy process on a
Euclidean space. On a Riemannian manifold, our definition (with Levi-Civita
connection) generalizes the Eells-Elworthy-Malliavin construction of the
Brownian motion and extends the class of isotropic L\'evy process introduced in
Applebaum and Estrade [AE00]. On a Lie group with a surjective exponential map,
our definition (with left-invariant connection) coincides with the classical
definition of a (left) L\'evy process given in terms of its increments.
  Our main theorem characterizes the class of L\'evy processes via their
generators on $M$, generalizing the fact that the Laplace-Beltrami operator
generates Brownian motion on a Riemannian manifold. Its proof requires a
path-wise construction of the stochastic horizontal lift and anti-development
of a discontinuous semimartingale, leading to a generalization of Pontier and
Estrade [PE92] to smooth manifolds with non-unique geodesics between distinct
points.
