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Since a few breakthroughs in the fundamental understanding of the effects of swift heavy ions (SHI) decelerating in the electronic stopping regime in the matter have been achieved in the last decade, it motivated us to review the state-of-the-art approaches in the modeling of SHI effects. The SHI track kinetics occurs via several well-separated stages: from attoseconds in ion-impact ionization depositing energy in a target, to femtoseconds of electron transport and hole cascades, to picoseconds of lattice excitation and response, to nanoseconds of atomic relaxation, and even longer macroscopic reaction. Each stage requires its own approaches for quantitative description. We discuss that understanding the links between the stages makes it possible to describe the entire track kinetics within a multiscale model without fitting procedures. The review focuses on the underlying physical mechanisms of each process, the dominant effects they produce, and the limitations of the existing approaches as well as various numerical techniques implementing these models. It provides an overview of ab-initio-based modeling of the evolution of the electronic properties; Monte Carlo simulations of nonequilibrium electronic transport; molecular dynamics modeling of atomic reaction on the surface and in the bulk; kinetic Mote Carlo of atomic defect kinetics; finite-difference methods of tracks interaction with chemical solvents describing etching kinetics. We outline the modern methods that couple these approaches into multiscale multidisciplinary models and point to their bottlenecks, strengths, and weaknesses. The analysis is accompanied by examples of important results improving the understanding of track formation in various materials. Summarizing the most recent advances in the field of the track formation process, the review delivers a comprehensive picture and detailed understanding of the phenomena.