Amplitudes predict the outcome of scattering experiments with particle colliders. Feynman's perturbative approach leads to considering a power series whose coefficients are computed by so-called Feynman integrals. The perturbative expansion of string theory amplitudes is an important testing ground for relations between gravity and gauge theories (double-copy relations, AdS/CFT). It is indexed by an integer which can be interpreted as the genus of a surface. In the last decade, the combined effort of mathematicians and physicists led to great progress in our understanding of the genus-zero and genus-one coefficients. I will report on this progress, with special focus on the role of Knizhnik-Zamolodchikov-Bernard (KZB) equations, which arise from Wess-Zumino-Witten models, and of polylogarithm functions. At the end I will report on higher-genus perspectives, based on a joint work with Benjamin Enriquez.