<p>This paper is mainly a survey of some recent developments and applications of advanced variational analysis to a class of controlled dynamical systems governed by discontinuous dissipative differential inclusions that are known as sweeping processes. Our first topic here concerns the well-posedness of sweeping processes in Hilbert spaces with controlled polyhedral moving sets. Then we formulate novel classes of optimal control problems for constrained sweeping dynamics with free time and provide necessary optimality conditions for local minimizers in the case of finite-dimensional state spaces. Our approach is based on new versions of the method of discrete approximations of their own interest with employing appropriate tools of first-order and second-order variational analysis and generalized differentiation.</p>