In the last years, Graph Convolutional Neural Networks gained popularity in the Machine Learning community for their capability of extracting local compositional features on signals defined on non-Euclidean domains. Shape correspondence, document classification, molecular properties predictions are just few of the many different problems where these techniques have been successfully applied. In this paper we will present Deep Geometric Matrix Completion, a recent application of Graph Convolutional Neural Networks to the matrix completion problem. We will illustrate MGCNN (a multi-graph CNN able to deal with signals defined over multiple domains) and we will show how coupling such technique with a RNN, a learnable diffusion process can be realized for reconstructing the desired information. Extensive experimental evaluation shows how Geometric Deep Learning techniques allow to outperform previous state of the art solutions on the matrix completion problem.