In this talk I will introduce the notion of conical end and (fibered) conic singularity for a Riemannian manifold and I will present
a Gauss-Bonnet formula in this context.
The deviation from the classical formula (proved in the 1940s by Allendoerfer, Weil and Chern) is a contribution for each end and each conical singularity of a term which is expressed as a sum of Lipschitz-Killing curvatures.