In this talk, we talk about the regularity of the free boundary in the Monge-Amp`ere obstacle problem \begin{equation}\begin{split}\det D^2 v= f(y)\chi_{{v>0}},\quad \text{in}\quad \Omega\v=v_0,\quad \text{on}\quad \partial\Omega.\end{split} \end{equation}Assume that $\Omega$ is a bounded convex domain in $\Bbb R^n$, and $f, v_0>0$.Then $\Gamma=\partial {v=0}$ is smooth if $f$ is smooth; and $\Gamma$ is analytic if $f$ is analytic. This is a joint work with Prof. Tang Lan and Prof. Wang Xu-Jia.