  We discuss the recent developments in the theory of diffusive shock
acceleration (DSA) by using both first-principle kinetic plasma simulations and
analytical theory based on the solution of the convection/diffusion equation.
In particular, we show how simulations reveal that the spectra of accelerated
particles are significantly steeper than the $E^{-2}$ predicted by the standard
theory of DSA for strong shocks, in agreement with several observational pieces
of evidence. We single out which standard assumptions of test-particle and
non-linear DSA are violated in the presence of strong (self-generated) magnetic
turbulence and put forward a novel theory in better agreement with the particle
spectra inferred with multi-wavelength observations of young SN remnants,
radio-supernovae, and Galactic cosmic rays.
