ABSTRACT Capture of interstellar objects (ISOs) into the Solar system is dominated by ISOs with asymptotic incoming speeds $v_\infty \lt 4\,$km s−1. The capture rate is proportional to the ISO phase-space density in the Solar vicinity and does not vary along the Sun’s Galactic orbit, i.e. is not enhanced during a passage through a cloud of ISOs (in contrast to previous suggestions). Most bound orbits crossing those of Jupiter and Saturn are fully mixed with unbound phase space, implying that they hold the same ISO phase-space density. Assuming an interstellar number density of ${n}_{\mathrm{iso}}\sim 0.1\,$au−3, we estimate that in 1000 years the planets capture ∼2 ISOs (while ∼17 fall into the Sun), resulting in a population of ∼8 captured ISOs within 5 au of the Sun at any time, less than the number of visiting ISOs passing through the same volume on hyperbolic orbits. In terms of phase-space volume, capture into and ejection from the Solar system are equal, such that on average ISOs will not remain captive at $a\lesssim 2000\,$au for extensive periods.