Abstract
In this lecture note I shall describe some recent developments of free boundary value problems and their regularity theory. Starting with the classical obstacle problem, and its development along with detailed proofs of each step of the theory, I shall go on to introduce various closely related semi-classical problems. The latter includes Bernoulli type FBP, Semi- and Quasi-linear PDEs with FB, Non-variational FB, and Non-local problems. As the techniques for treating the classical obstacle problem substantially differs from the others we have chosen to only discuss these problems from a heuristic point of view, and focus on approaches and the difficulties of these problems.
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