The author studies (set-valued) mappings of bounded \(\Phi\)-variation defined on the compact interval \(I\) and taking values in metric or normed linear spaces \(X\). He proves a new structural theorem for these mappings and extends Medvedev's criterion from real-valued functions onto mappings with values in a reflexive Banach space, which permits him to establish an explicit integral formula for the \(\Phi\)-variation of a metric space valued mapping. He shows that the linear span \(GV_\Phi(I; X)\) of the set of all mappings of bounded \(\Phi\)-variation is automatically a Banach algebra provided \(X\) is a Banach algebra. If \(h: I\times X\to Y\) is a given mapping and the composition operator \({\mathcal H}\) is defined by \(({\mathcal H}f)(t)= h(t, f(t))\), where \(t\in I\) and \(f: I\to X\), he shows that \({\mathcal H}: GV_\Phi(I; X)\to GV_\Psi(I; Y)\) is Lipschitzian if and only if \(h(t,x)= h_0(t)+ h_1(t)x\), \(t\in I\), \(x\in X\). This result is further extended to multivalued composition operators \({\mathcal H}\) with values in compact convex sets. Finally, he proves that any (not necessarily convex valued) multifunction of bounded \(\Phi\)-variation with respect to the Hausdorff metric, whose graph is compact, admits regular selections of bounded \(\Phi\)-variation.