We shall briefly present some recent results concerning S-integral points on certain varieties of dimension > 1. Some of them concern non-singular surfaces, some other ones arise from diophantine equations with linear recurrences. These results, obtained jointly with P. Corvaja, use the Schmidt Subspace Theorem applied to suitable auxiliary linear forms, constructed as values of certain regular functions on the variety. For the methods to apply, it is often necessary that the divisor at infinity is highly reducible; we give examples showing that on dropping this restriction very difficult problem arise even in simple contexts. We then show some instances when the reducibility condition may be achieved by taking an unramified cover, as in recent work by Faltings.
Generally speaking, we shall not assume prior knowledge of the subject and we shall often illustrate the theorems by quite concrete examples.