We consider piecewise-smooth dynamical systems, i.e., systems of ordinary differential equations switching between different sets of equations on distinct domains, separated by hyper-surfaces. As is well-known, when the solution approaches a discontinuity manifold, a classical solution may cease to exist. For this reason, starting with the pioneering work of Filippov, a concept of weak solution (also known as sliding mode) has been introduced and studied. Nowadays, the solution of piecewise-smooth dynamical systems in and close to discontinuity manifolds is well understood, if the manifold consists locally of a single discontinuity hyper-surface or of the intersection of two discontinuity hyper-surfaces. The present work presents partial results on the solution in and close to discontinuity manifolds of codimension 3 and higher.