Superstring amplitudes and multiple zeta values

The Feynman diagram expansion of scattering amplitudes in perturbative superstring theory can
be written as a series of integrals over compactified moduli spaces of Riemann surfaces with marked
points, indexed by the genus.
In genus 0 it is known that the amplitude can be expressed in terms of periods of M_{0,N}, which
are just multiple zeta values, by a theorem of Brown.
In this talk I want to report on recent advances in the genus 1 amplitude, which are related to the
development of 2 different generalizations of classical multiple zeta values, namely elliptic multiple
zeta values and conical sums.