Abstract: We consider computational aspects of using semidiscrete approximation schemes to solve problems with infinite-dimensional dynamics. We survey theory and convergence results for the simulation or forward problem, the feedback control problem, and the parameter estimation problem, all for the case in which the underlying system dynamics is governed by a partial differential equation. In particular we investigate the critical sufficient conditions required for convergence of semidiscrete approximations of these problems. These sufficient conditions require that the approximation scheme demonstrate system convergence, adjoint system convergence, and uniform preservation of system stability. By considering in detail several specific examples, we illustrate the difficulties which may arise when these sufficient conditions are not satisfied.