Quantum error correction is an essential prerequisite for fault-tolerant quantum comput- ing, and the surface code currently represents the most promising approach. Its deployment, however, critically depends on a decoder, the algorithm that, given the measured syndrome, infers the correction to be applied. The standard decoder, Minimum-Weight Perfect Match- ing, is fast and robust but suboptimal, as it decodes the X and Z error sectors separately and searches for the minimum-weight correction rather than the most statistically probable one, exploiting the fact that the two often coincide, yet remaining limited by the circumstance that this is not always the case. This thesis investigates neural decoders based on Graph Neu- ral Networks, with the objective of surpassing Minimum-Weight Perfect Matching in logical error rate, in a code capacity regime under depolarizing noise.