Abstract：

In this talk I will briefly present the historical development behind Chern's proof in 1944 of the higher dimensional Gauss-Bonnet formula, starting with the work of H. Hopf back in 1925. I will then introduce the notion of double forms (which basically are sums of tensor products of differential forms) and explain how these tensors help us reformulate the Gauss-Bonnet-Chern-Formula and in particular clarify the meaning of Chern's boundary term. I will give some application and if time permits, some perspectives on non compact and singular Riemannian manifolds.