On Schmidt's Subspace Theorem

Schmidt's Subspace Theorem is a higher dimensional generalization of Roth's Theorem on the approximation of algebraic numbers by elements from a given number field. It asserts that the set of solutions in $P^n(K)$ ($K$ number field) of a particular Diophantine inequality is contained in finitely many proper linear subspaces of $P^n(K)$. I would like to discuss quantitative versions giving an explicit upper bound for the number of subspaces, and go into ideas of Faltings that went into the proof.