In the early 80's, Yau conjectured that in any closed $3$-manifold there should be infinitely many minimal surfaces. I will review previous contributions to the question and present a proof of the conjecture, which builds on min-max methods developed by F. C. Marques and A. Neves. A key step is the construction by min-max theory of a sequence of closed minimal surfaces in a manifold N with non-empty stable boundary, and I will explain how to achieve this via the construction of a non-compact manifold with cylindrical ends.