We consider the classical Bernoulli percolation model on an infinite connected graph G. One of the most fundamental questions in the field is to understand for which graphs G the critical point for the emergence of an infinite connected component is non-trivial, i.e. $p_c(G)\in(0,1)$. While it is easy to prove that $p_c>0$ under very mild conditions, proving $p_c<1$ is a much harder problem. In fact, it is also easy to see that $p_c=1$ if $G$ has linear volume growth. In their seminal paper from 1996, Benjamini and Schramm conjectured that the converse is true for transitive graphs, i.e. $p_c(G)<1$ for every transitive graph G with superlinear volume growth. In this talk, we report on works proving this conjecture as well as a uniform upper bound for the case of Cayley graphs. The proofs rely on an interpolation scheme used to compare Bernoulli percolation with the excursion sets of the Gaussian Free Field, a strongly correlated percolation model for which one can easily prove the existence of a supercritical phase. Based on joint works with H. Duminil-Copin, S. Goswami, A. Raoufi, A. Yadin and C. Panagiotis.