Anomaly non-renormalization, lattice QFT and universality of transport coefficients

Recently new methods have been introduced to investigate the non-renormalization properties of the anomalies at a non perturbative level and in presence of a lattice. The issue is relevant in a number of problems ranging from the anomaly-free construction of chiral lattice gauge theory with large cut-off to the universality properties observed in transport coefficients in condensed matter systems. A review of main results and future perspectives is presented.