  We calculate the double resonant (DR) Raman spectrum of graphene, and
determine the lines associated to both phonon-defect processes, and two-phonons
ones. Phonon and electronic dispersions reproduce calculations based on density
functional theory corrected with GW. Electron-light, -phonon, and -defect
scattering matrix elements and the electronic linewidth are explicitly
calculated. Defect-induced processes are simulated by considering different
kind of idealized defects. For an excitation energy of $\epsilon_L=2.4$ eV, the
agreement with measurements is very good and calculations reproduce: the
relative intensities among phonon-defect or among two-phonon lines; the
measured small widths of the D, $D'$, 2D and $2D'$ lines; the line shapes; the
presence of small intensity lines in the 1800, 2000 cm$^{-1}$ range. We
determine how the spectra depend on the excitation energy, on the light
polarization, on the electronic linewidth, on the kind of defects and on their
concentration. According to the present findings, the intensity ratio between
the $2D'$ and 2D lines can be used to determine experimentally the electronic
linewidth. The intensity ratio between the $D$ and $D'$ lines depends on the
kind of model defect, suggesting that this ratio could possibly be used to
identify the kind of defects present in actual samples. Charged impurities
outside the graphene plane provide an almost undetectable contribution to the
Raman signal.
