  We present a new approach to metric Diophantine approximation on manifolds
based on the correspondence between approximation properties of numbers and
orbit properties of certain flows on homogeneous spaces. This approach yields a
new proof of a conjecture of Mahler, originally settled by V. Sprindzhuk in
1964. We also prove several related hypotheses of A. Baker and V. Sprindzhuk
formulated in 1970s. The core of the proof is a theorem which generalizes and
sharpens earlier results on non-divergence of unipotent flows on the space of
lattices.
