In this paper we present an extension of Moreau's sweeping process for higher order systems. The dynamical framework is carefully introduced, and preliminary well-posedness results are given. The time-discretisation of these nonsmooth systems with a time-stepping algorithm is also presented. This differential inclusion can be seen as a mathematical formulation of complementarity dynamical systems with arbitrary dimension and arbitrary relative degree between the complementary slackness variables. Applications of such high-order sweeping processes can be found in dynamic optimisation under state constraints and electrical circuits with ideal diodes.