In this talk, we consider the regularity of solutions and the regularity of the free boundary of the obstacle problems. Specifically, we study the regularity of the free boundary of a non-convex fully nonlinear operator and the regularity of solutions and the free boundary of the double obstacle problem. In order to prove the regularity of the free boundary of a non-convex fully nonlinear operator, we have the interior $C^2alpha$ regularity of the solution of the Dirichlet problem for the non-convex fully nonlinear operator. In the double obstacle problem for Laplacian, we use the ACF monotonicity formula and the Weiss` monotonicity formula. The monotonicity formulas are not applicable for the double obstacle problem for fully nonlinear operator. Hence, we exploit the fact that the term $partial_e u/x_n$ is finite, where $e$ is a direction orthogonal to $e_n$, for the global solution $u$ with the half space function type upper obstacle $psi=c(x_n^+)^2$.