In this seminar, I will present the problem of the propagation of singularities for differential equations and the corresponding problem of propagation of polarization for systems of differential equations. The problem has been solved for operators of principal type, but the theory is still incomplete for more general operators. Finally, I would like to present an example of an extremely degenerate operator that does not fall within the classical theory, yet for which propagation phenomena still occur. This example arises from differential geometry and concerns certain systems of PDEs related to the curvature tensors of a Riemannian or Lorentzian metric. N.B.: This series of talks is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006).