We present the well understood Stefan and Oxygen Depletion Problems as examples of explicit and implicit moving boundary value problems respectively. We show the different ways these problems can be analyzed and computed numerically with a focus on the Enthalpy Method (a grid capturing technique). Generalized problems are shown, including two phase flow in porous media and the formation of sea ice. Rigorous analysis of the problems and their numerical approximation is missing in many of these generalized problems.