The ``sweeping process'' was introduced by Moreau. In this problem, a convex closed ``moving'' set \(K(t)\) is considered, which ``drags'' a point \(x(t)\) that is constrained both to belong to the moving set for all \(t\), and also to have a velocity ``normal'' to the boundary of this set. The problem is formulated as follows: \[ -\dot x(t)\in N_{K(t)} \bigl(x(t)\bigr),\quad x(0)= x_0\in K(0), \quad x(t)\in K(t)\text{ for all }t, \] where \(N_{K(t)} (x(t))\) represents the normal cone to \(K(t)\). In the present article, the authors consider the problem in a Hilbert space in which \(K\) is not necessarily convex-valued. Because of this, the normal cone must be replaced by the Clarke normal cone in the formulation of the problem. Existence, uniqueness and regularity results are proven. In some existence theorems, for example, the convexity of \(K(t)\) is replaced by the weaker assumption of \(\varphi\)-convexity, extending results of Moreau.