Diophantine exponents of measures: a dynamical approach and submanifolds

We place the theory of metric Diophantine approximation on manifolds into a broader context of studying Diophantine properties of points generic with respect to certain measures on $\mathbb R^n$. The correspondence between multidimensional Diophantine approximation and dynamics of lattices in Euclidean spaces is discussed in an elementary way, and several recent results obtained by means of this correspondence are surveyed.