Motility-Induced Phase Separation (MIPS) is a phenomenon observed in self-propelled particles, such as bacteria and synthetic swimmers, where they tend to accumulate in regions of lower motility, leading to phase separation between dense and dilute fluid phases. This behavior arises from positive feedback between motility and density, where slower movement in high-density regions leads to further accumulation, and vice versa. The underlying mechanism can be understood through the master equation of self-propelled particles, which shows that their steady-state probability density is inversely proportional to their local speed. This effect is absent in passive Brownian particles, where the probability density depends only on temperature. MIPS can be mapped to equilibrium phase separation in certain limits, but breaks down at higher orders in gradients, leading to new effects without equilibrium counterparts. The study of MIPS involves both theoretical and numerical approaches, with simulations showing phase separation in run-and-tumble and active Brownian particles. The local approximation, where swim speed depends on density but not its gradients, is used to derive the dynamics of collective density and density-dependent motility parameters. However, beyond this approximation, nonlocal or gradient terms are necessary to capture the true dynamics of MIPS, which cannot be equivalent to any form of passive phase separation. Theoretical models show that MIPS can occur when the swim speed decreases with density, leading to a spinodal decomposition. This is confirmed by simulations of active particles in 1D and 2D, where phase separation is observed in both run-and-tumble and active Brownian particles. The phase diagrams of MIPS depend on the form of the swim speed as a function of density, with different behaviors for symmetric and asymmetric kernels. The study of MIPS also highlights the importance of non-equilibrium dynamics, as the system's steady states do not obey detailed balance, and the underlying non-equilibrium character cannot be transformed away. Numerical evidence for MIPS is supported by simulations of active particles in various potentials, showing phase separation in both hard-sphere and soft-sphere systems. The results indicate that MIPS occurs when the swim speed decreases with density, leading to a spinodal decomposition. The phase diagrams of MIPS depend on the form of the swim speed as a function of density, with different behaviors for symmetric and asymmetric kernels. The study of MIPS also highlights the importance of non-equilibrium dynamics, as the system's steady states do not obey detailed balance, and the underlying non-equilibrium character cannot be transformed away.Motility-Induced Phase Separation (MIPS) is a phenomenon observed in self-propelled particles, such as bacteria and synthetic swimmers, where they tend to accumulate in regions of lower motility, leading to phase separation between dense and dilute fluid phases. This behavior arises from positive feedback between motility and density, where slower movement in high-density regions leads to further accumulation, and vice versa. The underlying mechanism can be understood through the master equation of self-propelled particles, which shows that their steady-state probability density is inversely proportional to their local speed. This effect is absent in passive Brownian particles, where the probability density depends only on temperature. MIPS can be mapped to equilibrium phase separation in certain limits, but breaks down at higher orders in gradients, leading to new effects without equilibrium counterparts. The study of MIPS involves both theoretical and numerical approaches, with simulations showing phase separation in run-and-tumble and active Brownian particles. The local approximation, where swim speed depends on density but not its gradients, is used to derive the dynamics of collective density and density-dependent motility parameters. However, beyond this approximation, nonlocal or gradient terms are necessary to capture the true dynamics of MIPS, which cannot be equivalent to any form of passive phase separation. Theoretical models show that MIPS can occur when the swim speed decreases with density, leading to a spinodal decomposition. This is confirmed by simulations of active particles in 1D and 2D, where phase separation is observed in both run-and-tumble and active Brownian particles. The phase diagrams of MIPS depend on the form of the swim speed as a function of density, with different behaviors for symmetric and asymmetric kernels. The study of MIPS also highlights the importance of non-equilibrium dynamics, as the system's steady states do not obey detailed balance, and the underlying non-equilibrium character cannot be transformed away. Numerical evidence for MIPS is supported by simulations of active particles in various potentials, showing phase separation in both hard-sphere and soft-sphere systems. The results indicate that MIPS occurs when the swim speed decreases with density, leading to a spinodal decomposition. The phase diagrams of MIPS depend on the form of the swim speed as a function of density, with different behaviors for symmetric and asymmetric kernels. The study of MIPS also highlights the importance of non-equilibrium dynamics, as the system's steady states do not obey detailed balance, and the underlying non-equilibrium character cannot be transformed away.