A classical example of a non-degenerate elliptic obstacle problem arises when we consider a thin membrane in three-dimensional space suspended from a loop of wire, where the membrane is not free to assume a shape determined solely by gravity but rather is constrained to lie above a smooth surface or "obstacle", such as a sphere. A solution to the unconstrained thin membrane problem is obtained by solving a Poisson equation (with a Dirichlet boundary condition), whereas a solution to the constrained, obstacle problem is obtained by solving an obstacle problem defined by the Laplace operator. Obstacle problems of this kind, despite their apparent simplicity, are well known to give rise to difficult problems concerning the optimal (that is, C^{1,1}) regularity of the solution and, in particular, the regularity of the "free boundary" where the solution first meets the obstacle. Such regularity problems have been studied by Caffarelli, Shahgholian, and many others. In this lecture, we shall describe recent work with Panagiota Daskalopoulos and Camelia Pop, in which we extend previous optimal regularity results for solutions to obstacle problems defined by the Laplace operator to obstacle problems defined by degenerate-elliptic operators arising in stochastic analysis and other areas of applied mathematics, including mathematical finance.