Several rich mathematical objects appear in superstring scattering amplitudes, such as modular graph forms and Riemann zeta-values. Modular graph forms are modular forms associated with vacuum Feynman graphs. Both modular graph forms and zeta-values admit a grading by their transcendental weight. I shall review the transcendental structure of superstring amplitudes, including the conjectured uniform transcendentality of genus-one amplitudes in Type II superstring theory. I shall also describe recent work with Eric D’Hoker (arXiv:2110.06237) in which we evaluated the integrals of several infinite families of modular graph functions over genus-one moduli space.