We begin the tutorial by introducing a couple of interface problems in various physical and biological applications as motivating examples. We then present some basic ingredients of phase field models and take a special example to illustrate both analytical and computational aspects of the relevant variational problems and associated gradient dynamics. We discuss in particular dynamics related to the classical Allen-Cahn model and Cahn-Hilliard models with various mobilities. We next focus on a few phase field models for selected applications in biology and materials science which are developed for interface problems involving elastic energy contributions and those involving the coupling with external fields. Finally, we conclude the tutorial with discussions on other issues related to phase field modeling and simulations such as robust retrieval of statistical information, stochastic fluctuation and effective multiscale modeling.