An account is given of recent activity in the field of dynamics of phase separation and ordering processes in two-dimensional statistical mechanical models. The fundamental questions of the dynamics involve the form of the growth law, the value of the growth exponent, the dynamical scaling properties, and a possible universal classification of the late-stage dynamics. Evidence from kinetic lattice model calculations using computer-simulation techniques is presented in favor of a universal description of the dynamics in terms of algebraic growth laws with exponents which only depend on the nature of the conservation laws in effect. Atomic and molecular overlayers on solid surfaces and weakly-coupled atomic layers of certain three-dimensional crystals constitute a particularly suitable class of systems for studying fundamental aspects of ordering dynamics and phase separation in two dimensions since these systems provide a richness of ordering symmetries and degeneracies as well as they obey different conservation laws. Specific systems dealt with include the chemisorption systems O/W(110) and O/W(112), and oxygen layers in the basal CuO-planes of high-T(c) superconductors of the YBa2Cu3O7-delta-type.