We will discuss general features of the derivative expansion of Feynman diagrams contributions to the four-graviton scattering amplitudes in eleven-dimensional supergravity and type II superstring theories. This analysis will show that higher-loop amplitudes of maximally extended supergravity have a much better UV behaviour than expected and lead to various non renormalisation conditions on interactions of the form $D^{2k} R^4$ to type II superstring theories. We will discuss the range of validity of these results and what indications this provides on the role of supersymmetry in maximally supersymmetric theories of gravity. We will comment, as well, on the issue of the UV finitness of four-dimensional N=8 supergravity.

SPhT