One of the striking properties of a glassy system is that many material properties depend on its age, i.e., the time since the system entered its glassy phase. In this this talk we shall review some recent progress (work in collaboration with H. E. Castillo, P. Charbonneau, J. L. Iguain, M. P. Kennett, D. R. Reichman and M. Sellitto) in understanding local aging, through the study of local observable quantities, which reveal that there are spatial heterogeneities and fluctuations in the aging process of macroscopic systems. We show that a number of universal properties are shared by many non-equilibrium systems, both with and without quenched disorder, such as the 3D Edwards-Anderson model and some kinetically constrained non-interacting 2D and 3D spin models, for example.
Similar scaling relations are found for mesoscopic sample-to-sample fluctuations of global quantities in small size systems. We discuss how the emergence of a symmetry in aging systems, time-reparametrization invariance, could be responsible for the observed universal behavior of the local and mesoscopic non-equilibrium fluctuations.