The study of superstring scattering amplitudes provides a meeting ground for ideas from string theory, number theory, and algebraic geometry. Already at leading order in the perturbative expansion in the string coupling (tree level), one encounters rich arithmetic structures, such as single-valued multiple zeta values. At the next order, corresponding to one-loop corrections, both the analytic structure of the amplitudes and the techniques required to compute them become more intricate. In this talk, I will present recent progress on the one-loop computation of massless four- and five-point amplitudes in type IIB superstring theory.