In this course, we describe recent developments on minimal surface theory, with an emphasis on Morse-theoretic aspects. We talk about the Almgren-Pitts min-max theory of 1981, which can be used to construct a closed minimal hypersurface in a given compact Riemannian manifold. We overview the recent activity on the subject based on the use of multiparameter sweepouts, including the generic density of minimal hypersurfaces and other existence results.