The Kahler cone of a compact Kahler manifold is an open convex cone in a finite-dimensional real vector space which parametrizes the cohomology classes of Kahler 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 simplest nontrivial setting which is on K3 surfaces, and describe some recent results and open questions. This is joint work with Simion Filip.