In the calculation of the perturbative amplitude of superstring theory at one loop, modular graph functions (MGFs) emerge as notable mathematical constructs. These MGFs, representing Feynman diagrams on the surface of a torus, must be integrated over the fundamental domain. My talk will introduce MGFs, elucidate their generating series, and delve into the concept of equivariance, playing a key role in classifying MGFs. Additionally, I will cover recent advancements in understanding the generating series of MGFs and the integration of MGFs over the fundamental domain. The contents of this talk are derived from 2209.06772 and ongoing works.