Minimal hypersurfaces are higher dimensional analogues of geodesics. In the early 80's, S.-T. Yau conjectured that in any closed Riemannian 3-manifold, there exist infinitely many minimal surfaces. I will introduce the problem and give an account of recent developments, which led to the understanding that minimal hypersurfaces abound in closed Riemannian manifolds. In particular, Yau's conjecture is true and for generic metrics, much stronger properties hold: there are sequences of minimal hypersurfaces which equidistribute on average, and others that concentrate around a closed submanifold. I will discuss some ideas from the proofs, which borrow tools from analysis, geometric measure theory and topology. This talk is partially based on joint work with F.C. Marques, A. Neves and X. Zhou.