We will discuss some problems related to recovering information about an obstacle K in an Euclidean space from certain measurements of lengths of generalized geodesics in the exterior of the obstacle - e.g. sojourn times of scattering rays in the exterior of the obstacle, or simply travelling times of geodesics within a certain large ball containing the obstacle. It is well-known in scattering theory that this scattering data is related to the singularities of the scattering kernel of the scattering operator for the wave equation in the exterior of K with Dirichlet boundary condition on the boundary. It turns out that for some classes of obstacles, K can be completely recovered from the scattering data - we will describe some types of obstacles with this property. On the other hand, in general, obstacles cannot be completely recovered from scattering data. The impediment in such cases is the set of trapped points - when this set is too large, observability of the obstacle is impossible. We will discuss certain stability property of the trapping set - it turns out that the measure of the set of trapped points depends continuously on perturbations of the obstacle K. We will derive this from a certain generalisation of Santalo's formula to integrals over billiard trajectories (broken generalised geodesics) in the exterior of an obstacle. Some other applications of this formula to scattering by obstacles will be discussed as well.