During the last twenty years, inspired by the famous "BBM formula", many non-local characterizations of Sobolev and BV spaces have been studied in the literature. These characterizations usually rely on the limit (or the Gamma-limit) of suitable non-local functionals involving difference quotients instead of derivatives. In the first part of the talk, we present a simple proof of the "BBM formula", based on the sectioning technique, which provides both the pointwise and the Gamma-convergence. Then, in the second part of the talk, we describe some recent developments in this field. In particular we consider a general class of functionals introduced by Brezis, Seeger, Van Schaftingen and Yung, and we discuss some recent results and open questions related with these functionals.