We sketch a proof of a theorem due to Kleinbock, and generalizing previous work of Dani and of Margulis and Kleinbock, regarding the size of the set of bounded orbits of a mixing flow on a homogeneous space. We then discuss connections to number theory, specifically the fact, proved in the same paper of Kleinbock, that the set of badly approximable systems of affine forms has full Hausdorff dimension.