This talk will focus on some recent results concerning the low energy expansion of superstring scattering amplitudes. Whereas tree-level amplitudes generate a series of multiple zeta values (MZV) the genus-one amplitudes generate a series of elliptic generalisations (“modular graph functions”) that satisfy fascinating polynomial relations analogous to those satisfied by MZV’s. The latter part of the talk will briefly review how these features fit in with the non-perturbative structure of superstring amplitudes, studied some time ago.