The term moving interfaces and free boundary refers to a class of problems in which the domain of the problem itself is also an a priori unknown, as well as the basic unknown solution to governing equations. Hence, finding the domain is part of the problem. In this lecture I shall present some basic models in moving interfaces and free boundary problems. These includes obstacle problem, Stefan problem, Hele-Shaw flow and Muskat problem. I will also review the state of art for regularity results in obstacle problem.

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