Minimal surfaces are ubiquitous in geometry and applied science but their existence theory is rather mysterious. For instance, Yau in 1982 conjectured that any 3manifold admits infinitely many closed minimal surfaces but the best one knows is the existence of at least three. After a brief historical account, I will talk about my ongoing work with Marques and the progress we made on this question in our recent work with Irie and Song: we showed that for generic metrics, minimal surfaces are dense and equidistributed.
