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The lecture presents the results of theoretical, numerical and experimental investigations of combustion and detonation initiation in heterogeneous polydispersed mixtures. The problems of fuel droplets atomization, evaporation and combustion and the non-equilibrium effects in droplets atomization and phase transitions are discussed in details. The effects of droplets size non-uniformity and spatial distribution non-uniformity on mixture ignition and flame acceleration were investigated for strong and mild initiation of detonation: by a shock wave and spark ignition followed by deflagration to detonation transition (DDT). Peculiarities of jet injection and ignition in reaction chamber are studied. Investigating the behavior of individual droplets in a heated airflow allowed distinguishing two scenarios for droplet heating and evaporation. Small droplets undergo successively heating, then cooling due to heat losses for evaporation, and then rapid heating until the end of their lifetime. Larger droplets could directly be heated up to a critical temperature and then evaporate rapidly. Atomization of droplets interferes heating and evaporation scenario. Investigating droplet cloud strong ignition by a shock wave showed that increase of droplet concentration above definite value inhibits the onset of detonation in dispersed mixtures and gives birth to a detonation mode typical for non-premixed systems, when chemical reaction takes place only in a thin border layer of the cloud thus supporting the shock wave in pure gas propagation.