Minimal surfaces are among the most natural objects in Differential Geometry, and are fundamental tools in the solution of several important problems in mathematics. In these two lectures we will discuss the variational theory of minimal surfaces  and describe recent applications to geometry and topology, as well as mention some future directions in the field. 

In particular we will discuss our joint work with Andre Neves on the min-max theory for the area functional. This includes the solution of the Willmore conjecture and the construction of infinitely many minimal hypersurfaces in manifolds with positive Ricci curvature. We will also discuss joint work with Agol and Neves on the Freedman-He-Wang conjecture about links.