Galactic cosmic rays are believed to be accelerated at supernova remnant shocks. We model the dynamical evolution of type Ia and type II supernova remnants, and discuss how the maximum energy of accelerated protons and electrons is expected to evolve in time. In particular, we discuss whether supernova remnants can accelerate protons all the way up to PeV energies, and the crucial role of of magnetic field amplification in this context. The efficiency of field amplification is modeled in a phenomenological way, but some theoretical frameworks that may explain this mechanism are examined (in particular, the non-resonant, current-driven instability proposed by Bell), with emphasis on the role of the slope of the CR spectrum.In the final part we present the predicted evolution of the SNR gamma-ray spectrum for two benchmark cases of SNR I and II, we discuss in detail the impact of several crucial parameters (gas density, magnetic field inside the remnant, structure of the interstellar medium surrounding the SNR), and eventually compare our predictions with current observations of young SNRs in our Galaxy.