  Scattering amplitudes which describe the interaction of physical states play
an important role in determining physical observables. In string theory the
physical states are given by vibrations of open and closed strings and their
interactions are described (at the leading order in perturbation theory) by a
world-sheet given by the topology of a disk or sphere, respectively.
  Formally, for scattering of N strings this leads to N-3-dimensional iterated
real integrals along the compactified real axis or N-3-dimensional complex
sphere integrals, respectively. As a consequence the physical observables are
described by periods on M_{0,N} - the moduli space of Riemann spheres of N
ordered marked points.
  The mathematical structure of these string amplitudes share many recent
advances in arithmetic algebraic geometry and number theory like multiple zeta
values, single-valued multiple zeta values, Drinfeld, Deligne associators, Hopf
algebra and Lie algebra structures related to Grothendiecks Galois theory. We
review these results, with emphasis on a beautiful link between generalized
hypergeometric functions describing the real iterated integrals on M_{0,N}(R)
and the decomposition of motivic multiple zeta values. Furthermore, a relation
expressing complex integrals on M_{0,N}(C) as single-valued projection of
iterated real integrals on M_{0,N}(R) is exhibited.
