Proper Orthogonal Decomposition (POD) allows to compress information by identifying the most energetic modes obtained from a database of snapshots. In this work, POD is used to predict the behavior of compressible flows by means of global and local approaches which exploit some features of a discontinuous Galerkin spatial discretization. The presented global approach requires the definition of high-order and low-order POD bases which are built from a database of high-fidelity simulations. Predictions are obtained by performing a cheap low-order simulation whose solution is projected on the low-order basis. The projection coefficients are then used for the reconstruction with the high-order basis. However, the non-linear behavior related to the advection term of the governing equations makes the use of global POD bases quite problematic. For this reason, a second approach is presented in which an empirical POD basis is defined in each element of the mesh. This local approach is more intrusive with respect to the global approach but it is able to capture better the non-linearities related to advection. The two approaches are tested and compared on the inviscid compressible flow around a gas-turbine cascade and on the compressible turbulent flow around a wind turbine airfoil.