We give a survey of some recent regularity results for linear evolutionary partial differential equations and oscillatory integral operators. The emphasis will be on those oscillatory integral operators that are involved in proving continuous dependence on the initial data (for fixed time) for evolutionary PDEs. The results are concerned with regularity in classical function spaces, i.e. Besov-Lipschitz and Triebel-Lizorkin spaces in both Banach and quasi-Banach scales.