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.