<small></small>In a project with P. Balint, J. De Simoi and V. Kaloshin, we have studied the inverse problem for a class of open dispersing billiards satisfying a non-eclipse condition. The dynamics of such billiards is of type Axiom A and can be coded symbolically in a natural way, which allows us to define a marked length spectrum (lengths of all periodic orbits + coding). We have shown that this dynamical spectrum contains a lot of geometric information; in particular, when the billiard table has analytic boundary and a partial Z_2\times Z_2 symmetry, it is generically possible to determine the geometry of the table from its marked length spectrum. In a joint work with A. Florio, we have carried on with the study of such billiards in the case where the boundary is merely C^k. We show that open dispersing billiards satisfying the non-eclipse condition are spectrally rigid in the following sense: two such billiards which have the same marked length spectrum share the same geometry at the points of the table ÂŤ seen Âť by periodic orbits, i.e., on the projection of the Cantor set of trapped orbits on the boundary of the table. In particular, when the boundary of these tables is (quasi-)analytic, this implies that the two tables are isometric. One step of the proof is a general dynamical result on smooth conjugacy classes of 3D contact Axiom A flows, which extends a previous result due to J. Feldman and D. Ornstein in the Anosov case. Some ideas of our work are inspired by the techniques introduced by J.-P. Otal in his work on the spectral rigidity of geodesic flows on negatively curved surfaces.