This article is largely concerned with the relationships between the implicit (backward) discretization of set-valued sliding-mode systems (for control, state observation, or differentiation), and proximal-point algorithms of optimization. It is shown that using well-known notions like maximal monotonicity, proximal and resolvent operators, and Yosida approximations, it is possible to embed implicit time-discretization methods into the unified framework of proximal-point algorithms. This is illustrated with first-order and higher-order sliding-mode systems. Passivity is shown to play an important role. Robust, time-varying, Hamiltonian and high-order extensions of the classical proximal-point algorithm are introduced. Particular attention is given to the use of splitting methods to resolve computational issues related to implementation.