In 2002, Corvaja and Zannier obtained a new proof of Siegel's theorem (on integral points on curves) based on Schmidt's celebrated Subspace Theorem. Soon after that (and based on earlier work), Evertse and Ferretti applied Schmidt's theorem to give diophantine results for homogeneous polynomials of higher degree on a projective variety in Pn. This has led to further work of A. Levin, P. Autissier, M. Ru, G. Heier, and others. In particular, Ru has defined a number, \Nev(D), that concisely describes the best diophantine approximation obtained by this method, where D is an effective Cartier divisor on a projective variety X. In this talk, I will give an overview of variants of \Nev(D) as developed by Ru and myself, and indicate how an example of Faltings can be derived using these constants.