We discover a new contribution to the pressure (or stress) exerted by a suspension of self-propelled bodies. Through their self-motion, all active matter systems generate a unique swim pressure that is entirely athermal in origin. The origin of the swim pressure is based upon the notion that an active body would swim away in space unless confined by boundaries – this confinement pressure is precisely the swim pressure. Here we give the micromechanical basis for the swim stress and use this new perspective to study self-assembly and phase separation in active soft matter. The swim pressure gives rise to a nonequilibrium equation of state for active matter with pressure-volume phase diagrams that resemble a van der Waals loop from equilibrium gas-liquid coexistence. Theoretical predictions are corroborated by Brownian dynamics simulations. Our new swim stress perspective can help analyze and exploit a wide class of active soft matter, from swimming bacteria to catalytic nanobots to molecular motors that activate the cellular cytoskeleton.