Existence of infinitely many minimal hypersurfaces in closed manifolds

In the early 80's, Yau conjectured that in any closed $3$-manifold there should be infinitely many closed minimal surfaces. I will survey previous results related to the question and present a proof of the conjecture. It builds on the min-max theory of F. Almgren and J. Pitts, which has recently been further developed by F. C. Marques and A. Neves.