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The physical picture of the phenomenon of high-power femtosecond laser pulse filamentation in air is presented. The report starts with short flashback to discovery of a laser beam self-focusing which underlies pulse filamentation. Physical parameters of filaments and their properties are reviewed. The main part focuses on laser pulse filamentation in ambient air. Pulse filamentation in the turbulent atmosphere and its study by means of computer simulations are discussed. Filaments can generate on stochastic perturbations caused by the refractive index fluctuations in turbulent atmosphere and scattering on aerosol particles. We investigated numerically multifilamentation of femtosecond laser pulse in atmosphere using the random phase screen model of light propagation in randomly inhomogeneous medium, based on the split-step method. At long distances in the atmosphere turbulent fluctuations of refractive index randomly change filamentation onset both along propagation path and in the beam cross-section. We discuss pointing stability and longitudinal wandering of filaments. There are two antithetical tendencies in filament formation in turbulent atmosphere. These are the increase of filament start distance under relatively low pulse power when one can see the self-focusing of the bulk pulse and the decrease of this distance, when and modulation instability develops, if the peak power of the pulse is much greater than critical power for self-focusing.