I will report on a series of joint papers with Gilles Carron (Nantes Université) and Ilaria Mondello (Université Paris-Est Créteil), in which we investigate the geometric and analytic properties of Kato limit spaces. These are defined as Gromov-Hausdorff (GH) limits of complete Riemannian manifolds satisfying uniform Kato bounds on the Ricci curvature. After reviewing these Kato bounds, I will provide a simple 2D example of a branching Kato limit space that cannot be GH-approximated by manifolds with a uniform Ricci curvature lower bound. I will also explain why this example does not lie in the GH closure of the set of manifolds satisfying a small Ricci curvature Lp bound in the sense of Petersen-Wei.