The Schmidt Subspace Theorem, its S-arithmetic extension by Schlickewei, and subsequent (rather significant) refinements are highlights of the theory of Diophantine applications and have many applications, some quite unexpected. The line of research that led to this remarkable theorem started with Thue (or even Liouville), and is closely related to Roth famous theorem about how well one can approximate irrational algebraic numbers by rational numbers --- more precisely, Roth tells us that for any irrational algebraic number alpha and epsilon >0, there are only finitely many rational numbers p/q such that |alpha - p/q|<q^{-2-epsilon}.

I will present the Subspace Theorem and some elements of its proof as well as some applications. The relation between this theorem and the theory of flows on homogeneous spaces (particularly, the actions of the diagonal group on the space of lattices in R^n) will also be explained.