We describe the basic modeling, analysis and approximation tools for phase transition models which use monotone multi-valued graphs. The PDE models with these feature free boundaries and lack of smoothness. We connect these to real-life applications of solid-liquid and soild-vapor p[hase transitions which are driven by, respectively, temperature and concentration gradients. We also discuss models where these are coupled. This is based on joint work with students and collaborators who will be named in the talk.