Given a Lie group G and a lattice \Gamma in G, we consider a flow on G/\Gamma induced by the action of a one-parameter subgroup of G. If this flow is mixing then a generic orbit is dense, but nevertheless one can discuss the dimension of the set of exceptions. We discuss work of S. G. Dani, in which such estimates are made in certain cases and a connection to diophantine approximation is established, and also generalizations due to D. Kleinbock and G. Margulis. In particular, we outline a dynamical proof, due to Kleinbock, that the set of badly approximable systems of affine forms has full Hausdorff dimension.