Abstract: Intermediate k^th Ricci curvatures are curvature conditions interpolating between Sectional curvature (k=1) and Ricci curvature(k=n-1). In this talk I will give a broad overview of what is and isn't known or expected about spaces admitting such metrics, on both sides of the apparent behavioral breakpoint of k=n/2. As an example, I will sketch the proof of an upcoming result that spaces with positive Ric_2 and some fixed degree of symmetry (say an action by T^10) must satisfy the Hopf conjecture, i.e. have positive Euler characteristic, and that the possible cohomology of the fixed points are very restricted. This is joint work with Lee Kennard and Lawrence Mouillé