The contributions contained in the volume, written by leading experts in their respective fields, are expanded versions of talks given at the INDAM Workshop Anomalies in Partial Differential Equations held in September 2019 at the Istituto Nazionale di Alta Matematica, Dipartimento di Matematica Guido Castelnuovo , Universita di Roma La Sapienza . The volume contains results for well-posedness and local solvability for linear models with low regular coefficients. Moreover, nonlinear dispersive models (damped waves, p-evolution models) are discussed from the point of view of critical exponents, blow-up phenomena or decay estimates for Sobolev solutions. Some contributions are devoted to models from applications as traffic flows, Einstein-Euler systems or stochastic PDEs as well. Finally, several contributions from Harmonic and Time-Frequency Analysis, in which the authors are interested in the action of localizing operators or the description of wave front sets, complete the volume.