We give necessary and sufficient conditions guaranteeing that the coupling
for L\'evy processes (with non-degenerate jump part) is successful. Our method
relies on explicit formulae for the transition semigroup of a compound Poisson
process and earlier results by Mineka and Lindvall-Rogers on couplings of
random walks. In particular, we obtain that a L\'{e}vy process admits a
successful coupling, if it is a strong Feller process or if the L\'evy (jump)
measure has an absolutely continuous component.Comment: 14 page
