A Riemannian metric on a closed manifold is called Zoll when all of its geodesics are closed and have the same period. An infinite dimensional family of Zoll metrics on the two-dimensional sphere were constructed by Otto Zoll in the beginning of 1900's, but many questions about them remain unanswered. In this talk, I will explain my motivation to look for higher dimensional analogues of Zoll metrics, where closed geodesics are replaced by embedded minimal spheres of codimension one. Then, I will discuss some recent results about the construction and geometric understanding of these new geometries. This is a joint project with F. Marques (Princeton) and A. Neves (UChicago).