In the last decade, the study of pressure in active matter has attracted growing attention due to its fundamental relevance to nonequilibrium statistical physics. Active matter systems are composed of particles that consume energy to sustain persistent motion, which are inherently far from equilibrium. These particles can exhibit complex behaviors, including motility-induced phase separation, density-dependent clustering, and anomalous stress distributions, motivating the introduction of active swim stress and swim pressure. Unlike in passive fluids, pressure in active systems emerges from momentum flux originated from swim force rather than equilibrium conservative interactions, offering a distinct perspective for understanding their mechanical response. Simple models of active Brownian particles (ABPs) have been employed in theoretical and simulation studies across both dilute and dense regimes, revealing that pressure is a state function and exhibits a nontrivial dependence on density. Together with nonequilibrium statistical concepts such as effective temperature and effective adhesion, pressure offers important insight for understanding behaviors in active matter such as sedimentation equilibrium and motility induced phase separation. Extensions of ABPs models beyond their simplest form have underscored the fragility of pressure-based equation of state, which can break down under factors such as density-dependent velocity, torque, complex boundary geometries and interactions. Building on these developments, this review provides a comprehensive survey of theoretical and experimental advances, with particular emphasis on the microscopic origins of active pressure and the mechanisms underlying the breakdown of the equation of state.

17 pages, 10 figures