The idea of finding closed geodesics in surfaces  by using sweep-outs goes back to the work of Birkhoff in the 1920s. Minimal surfaces in three-manifolds can also be constructed that way. This was the main accomplishment of Almgren and Pitts (1981), using powerful tools of Geometric Measure Theory. In this talk I will describe some unexpected relations between the min-max approach for minimal surfaces and conformally invariant variational problems in Geometry and Topology. More concretely, I will discuss the theory developed In a joint paper with Andre Neves, in which we prove the  Willmore conjecture (1965) by doing min-max to  a nontrivial 5-cycle in the space of surfaces in the three-sphere. I will also discuss a joint paper with Ian Agol and Andre Neves, in which we  prove the Freedman-He-Wang conjecture (1994) about the energy of links in three-space using similar methods.