The classical Jacobi equation is a linearization of the geodesic equation, and its solutions give topological and geometrical information about the manifold. A few years ago, Burkhard Wilking found a far reaching generalization of it, the transverse Jacobi equation, whose geometric significance is still far from understood. In this talk, aimed for the general public, we will show how to do some comparison geometry using it, and will try to give an idea of why the right context for the new comparison theorems is the notion of intermediate Ricci curvature.