When irradiating crystalline insulators by strong and short laser pulses the non-linear response of the material gives rise to high harmonic generation (HHG). In view of compact solid-state attosecond sources and electronics at optical frequencies there is great interest from both experimentalists and theorists in understanding and explaining the underlying physical process which, however, has not been achieved so far. We perform ab-initio simulations based on time dependent density functional theory (TDDFT) describing the ultrafast electron dynamics. In a methodological section we describe a newly developed numerical scheme to include damping (energy loss of excited electrons) and dephasing (decoherence) into TDDFT. The e ects of such processes on the emission of high harmonics are discussed using a (numerically) \simple" target material, diamond. Then, we focus on the simulation of two recent experiments. We study directional dependence and relative weights of the occurring inter- and intraband HHG from SiO2 irradiated by transients (sub-cycle pulses at optical frequencies) and we investigate the modi cations of HHG spectra due to the presence of an initial excited carrier population which was motivated by pump-probe experiments on HHG from ZnSe.