Physical aging is a ubiquitous phenomenon in glassy materials and originates from the fact that they are generally out-of-equilibrium. Due to the technological and fundamental implications, this phenomenon has been deeply investigated in the last decades especially in glassy polymers. Here we provide a critical review of the latest hot debated themes in the field of physical aging in polymers and polymer nanocomposites. We first summarize the fundamental aspects of physical aging, highlighting its relationship with the polymer segmental mobility. A review of the methods employed to monitor physical aging is also provided, in particular those probing the time dependent evolution of thermodynamic variables (or related to) and those probing the (quasi)instantaneous polymer segmental mobility. We subsequently focus our attention on the two following debated topics in the field of physical aging of polymers: (i) the fate of the dynamics and thermodynamics of glassy polymers below the glass transition temperature (T_g), i.e. the temperature below which physical aging occurs; (ii) the modification of physical aging induced by the presence of inorganic nanofillers in polymer nanocomposites. With respect to the former point particular attention is devoted to recent findings concerning possible deviations from the behavior normally observed above T_g of both dynamics and thermodynamics deep in the glassy state. Regarding the effect of the presence of nanofillers on the rate of physical aging, the role of the modification of the polymer segmental mobility and that of purely geometric factors are discussed with particular emphasis on the most recent advances in the topic. The modification of the rate of physical aging in other nanostructured systems, such as polymer thin films, is discussed with particular emphasis on the analogy in terms of a large amount of interface with polymer nanocomposites.