Perturbative finiteness of superstring theory

In these proceedings, the multiloop amplitude prescription using the super-Poincaré invariant pure spinor formalism for the superstring is reviewed. Unlike the RNS prescription, there is no sum over spin structures and surface terms coming from the boundary of moduli space can be ignored. Massless N-point multiloop amplitudes vanish for N 4, which implies (with two mild assumptions) the perturbative finiteness of superstring theory. And R4 terms receive no multiloop contributions in agreement with the Type IIB S-duality conjecture of Green and Gutperle.