The paper by Cates and Tailleur explores the phenomenon of Motility-Induced Phase Separation (MIPS) in self-propelled particles, which include both synthetic colloids and microorganisms. Self-propelled particles, such as bacteria and active Brownian particles, exhibit unique behaviors due to their continuous energy consumption, leading to deviations from equilibrium thermodynamics. The authors discuss how these particles tend to accumulate where they move more slowly, and how this can result in density-dependent slowing, creating a positive feedback loop that leads to MIPS between dense and dilute fluid phases. The paper provides a theoretical framework for understanding MIPS, including the mapping of variable-speed self-propelled particles to passive particles with attractions. This mapping is confirmed by simulations but breaks down at higher gradients, leading to new effects not present in equilibrium systems. The authors also review the experimental implications of MIPS, highlighting its potential applications in controlling the assembly of nanostructures and understanding biofilm formation. Key aspects of MIPS include the dynamics of collective density, the role of density-dependent motility parameters, and the local approximation where the swim speed depends only on the density. The paper discusses the conditions for MIPS, such as the linear instability when the derivative of the swim speed with respect to density is less than a critical value. It also explores the effects of finite thermal diffusivity and the role of gradient terms in the dynamics. Numerical evidence for MIPS is presented through simulations of run-and-tumble particles and active Brownian particles, showing that MIPS occurs when the swim speed decreases sufficiently with increasing density. The paper concludes with a discussion of the broader implications of MIPS and its potential for future research.The paper by Cates and Tailleur explores the phenomenon of Motility-Induced Phase Separation (MIPS) in self-propelled particles, which include both synthetic colloids and microorganisms. Self-propelled particles, such as bacteria and active Brownian particles, exhibit unique behaviors due to their continuous energy consumption, leading to deviations from equilibrium thermodynamics. The authors discuss how these particles tend to accumulate where they move more slowly, and how this can result in density-dependent slowing, creating a positive feedback loop that leads to MIPS between dense and dilute fluid phases. The paper provides a theoretical framework for understanding MIPS, including the mapping of variable-speed self-propelled particles to passive particles with attractions. This mapping is confirmed by simulations but breaks down at higher gradients, leading to new effects not present in equilibrium systems. The authors also review the experimental implications of MIPS, highlighting its potential applications in controlling the assembly of nanostructures and understanding biofilm formation. Key aspects of MIPS include the dynamics of collective density, the role of density-dependent motility parameters, and the local approximation where the swim speed depends only on the density. The paper discusses the conditions for MIPS, such as the linear instability when the derivative of the swim speed with respect to density is less than a critical value. It also explores the effects of finite thermal diffusivity and the role of gradient terms in the dynamics. Numerical evidence for MIPS is presented through simulations of run-and-tumble particles and active Brownian particles, showing that MIPS occurs when the swim speed decreases sufficiently with increasing density. The paper concludes with a discussion of the broader implications of MIPS and its potential for future research.