In this talk we will present an overview of regularity results (both for the solution and for the free boundary) in a class of problems which arises in permeability theory. We will mostly focus on the parabolic Signorini (or thin obstacle) problem, and discuss the modern approach to this classical problem, based on several families of monotonicity formulas. In particular, we will present the optimal regularity of the solution, the classi?cation of free boundary points, the regularity of the regular set, and the structure of the singular set. These results have been obtained in joint work with N. Garofalo, A. Petrosyan, and T. To. We will also discuss the regularity of solutions in a related model arising in problems of semi-permeable walls and of temperature control. This is joint work with T. Backing.