We will describe a recently developed general approach to the
problem of intrinsic Diophantine approximation on homogeneous
algebraic varieties, raised by Serge Lang in 1965, and demonstrate
it in many natural examples. The approach utilizes harmonic analysis
in the automorphic representation associated with a lattice in an
algebraic group, together with some arguments in homogeneous dynamics
and ergodic theory. It provides the best possible solution to some
previously inaccessible intrinsic Diophantine approximation problems,
and raises some challenging open problems in others.

Based on joint work with Anish Ghosh and Alex Gorodnik.