Inverse obstacle scattering is the recovery of the boundary of a homogeneous object given scattering measurements far from the object. This can be contrasted with inverse medium scattering where some continuously varying (and compact) material parameter is to be recovered. The obstacle recovery problem has the benefit that the governing PDEs are generally homogeneous boundary value problems; standard boundary integral representations reduce the dimension of the PDE discretization by one. However, the obstacle setting introduces some difficulties. The space of solutions is non-convex and the natural regularizations of the problem are non-linear. In this talk, we’ll review some numerical methods for the obstacle problem that utilize ideas originally applied to the medium problem by Yu Chen. Then we’ll present some numerical methods and results of two recent papers that concern the obstacle scattering problem for trapping and dissipative media, respectively, and discuss some remaining challenges. This work is in collaboration with Carlos Borges, Jeremy Hoskins, and Manas Rachh.