We will describe a recently developed general approach to Diophantine approximation on homogeneous algebraic varieties, and demonstrate it in some familiar natural examples. The approach utilizes harmonic analysis on Lie groups and some arguments in homogeneous dynamics and ergodic theory. It provides  the best possible solution to a host of intrinsic Diophantine approximation problems which were not accessible by previous techniques. Based on joint work with Alex Gorodnik and on joint work with Anish Ghosh and Alex Gorodnik.