The cone of nef (1,1) real Bott-Chern cohomology classes on a compact complex manifold is defined in analogy with the Kahler case, as the cone of classes which admit representatives with arbitrarily small negative part. On a K\"ahler manifold this coincides with the closure of the Kahler cone. I will review some basic open problems about these classes, which are either trivial or known in the Kahler case, and I will present partial progress towards some of these (joint with S. Kolodziej and B. Weinkove) and some speculations.