  The $(2k)$-th Gauss-Bonnet curvature is a generalization to higher dimensions
of the $(2k)$-dimensional Gauss-Bonnet integrand, it coincides with the usual
scalar curvature for $k=1$. The Gauss-Bonnet curvatures are used in theoretical
physics to describe gravity in higher dimensional space times where they are
known as the Lagrangian of Lovelock gravity, Gauss-Bonnet Gravity and Lanczos
gravity. In this paper we present various aspects of these curvature invariants
and review their variational properties. In particular, we discuss natural
generalizations of the Yamabe problem, Einstein metrics and minimal
submanifolds.
