<p>This paper addresses optimal control problems for dynamical systems governed by a novel class of sweeping processes with time delay. We establish the well-posedness of such processes in the sense of the existence and uniqueness of feasible trajectories corresponding to feasible controls under fairly unrestrictive assumptions. We then construct a well-posed family of discrete approximations and find efficient conditions ensuring the stability of the discretized time-delayed sweeping process with respect to strong convergence of feasible and optimal solutions. This creates a bridge between the optimization of continuous-time and discrete-time sweeping control systems and justifies the effective use of discrete approximations in deriving optimality conditions and numerical techniques to solve the original time-delayed sweeping control problems via discrete approximations.</p>