A summary of recent numerical simulations of ordering in several two-dimensional models is presented. Primary emphasis is given to simulations of domain growth in the kinetic Ising, Langevin and cell-dynamics models with conserved order parameters. The modified Lifshitz-Slyozov-Wagner domain-growth law is found to be an excellent fit to the data in all cases. In addition, the nonequilibrium pair correlation functions are shown to satisfy the same universal scaling form. The structure factors are also shown to satisfy dynamical scaling. Thus all three models are shown to belong to the same dynamical universality class, for the range of space and time considered in these studies.