“We present an approach that allows one to introduce a Malliavin type calculus for functionals of general Lévy processes and to obtain sufficient conditions for the absolute continuity of solutions of stochastic differential equations with jumps (we do not pose any assumptions about regularity of the intensity of the jumps). Our investigations are motivated by a pioneering idea due to Bismut ... and developed further by many authors. The idea is to extend the Malliavin approach to regularity of Wiener functionals to more general probability spaces by introducing a smooth structure in these spaces in terms of a “differentiation rule”, integration-by-parts formula, and by further applications of the stochastic calculus of variations to smooth functionals with nondegenerate derivatives.”