Abstract: The Kähler cone of a compact Kähler manifold is an open convex cone in a finite-dimensional real vector space which parametrizes the cohomology classes of Kähler metrics. A very natural problem is to understand the nature of its boundary, and whether cohomology classes on the boundary have representatives which have geometric significance. I will discuss in detail the case of Calabi-Yau manifolds, where these boundary classes can be probed using Ricci-flat metrics, and describe some recent results and open questions.