The dynamics of far-from-equilibrium ordering processes in multi-component systems, e.g. crystalline solids or fluid mixtures, that are being quenched in temperature is studied by computer-simulation techniques involving Monte Carlo (MC) as well as molecular dynamics (MD) methods. The systems are modeled by using simple two-dimensional statistical mechanical models such as multi-state Potts lattice models and models of particle systems interacting via Lennard-Jones-like potentials. The ordering dynamics is investigated under the conditions of both conservation and non-conservation of the order parameter as well as with and without the presence of additional 'foreign' components, such as vacancies and surfactants, that couple to the interfaces which develop during the ordering process. The present paper reviews recent progress in this field with an emphasis on a possible universal description of ordering dynamics.