High harmonic generation (HHG) is an extreme non-linear phenomenon where strong laser fields interact with a medium to produce coherent and high-frequency harmonics of the incident light. It has emerged as a rapidly growing research area in bulk materials since its first observation in ZnO crystals in 2011. Over the past decade, pioneering studies have already been made in understanding the details of the microscopic mechanism behind this phenomenon, like the role of intra- and inter-band transitions, the contribution of the modulus and the phase of the dipole moment to even and odd harmonic peaks, the role of the oscillating dipoles, effects of broken symmetry, etc. However, the role of electron-electron correlations in the HHG from strongly correlated materials is much less understood. In these materials the interactions between electrons play a significant role, leading to complex and intriguing physical behaviors. In this dissertation, on the example of ZnO, perovskites BaTiO3 and BiFeO3, and transition-metal oxide VO2 I will study the role of electron-electron interaction effects in the HH spectra by using the time-dependent density-functional theory (TDDFT) approach with the exchange-correlation kernel obtained with dynamical mean- field theory (DMFT). In DMFT, one takes into account time-resolved on-site electron-electron interactions (neglected in most of other approaches) that are crucial for a larger part of strongly correlated materials. As I demonstrate, correlation effects significantly modify the HH spectrum, e.g., through the ultrafast modification of the spectrum of the system, as it was found for ZnO. As the next step, I explored the effects of electron-electron correlations in the HH spectrum of BaTiO3 perturbed by intense, few-cycle mid-infrared laser excitations. The correlation effects in this system lead to the emergence of "super-harmonics" - periodic enhancements and suppressions of specific harmonic orders that depend on the correlation strength. I extended my analysis to the case of BiFeO3, where in addition to correlation effects the effects of memory in HHG were analyzed. I have found that both correlation effects and memory lead to an extension of the harmonic cutoff. In my final part, I explored the effect of electron correlations on the HH spectrum of VO2 and compared my findings with the experiment. The obtained results may shed light on the often important role of electron correlations in the HH spectra of solids, providing valuable insights into ultrafast dynamics in complex materials, and contributing to advancements in nonlinear optics and strong-field physics, with the potential for novel photonic devices and imaging techniques in the attosecond and femtosecond regimes.