We conduct a systematic survey of the regions in which distant satellites can orbit stably around the four giant planets in the solar system, using orbital integrations of up to 10 9 yr. In contrast to previous investigations, we use a grid of initial conditions on a surface of section to explore phase space uniformly inside and outside the planet's Hill sphere (radius r H ; satellites outside the Hill sphere sometimes are also known as quasi-satellites). Our confirmations and extensions of old results and new findings include the following: (1) many prograde and retrograde satellites can survive out to radii ∼0.5r H and ∼0.7r H , respectively, while some coplanar retrograde satellites of Jupiter and Neptune can survive out to ∼r H ; (2) stable orbits do not exist within the Hill sphere at high ecliptic inclinations when the semimajor axis is large enough that the solar tide is the dominant non-Keplerian perturbation; (3) there is a gap between ∼r H and 2r H in which no stable orbits exist; (4) at distances 2r H stable satellite orbits exist around Jupiter, Uranus, and Neptune (but not Saturn). For Uranus and Neptune, in particular, stable orbits are found at distances as large as ∼10r H ; (5) the differences in the stable zones beyond the Hill sphere arise mainly from differences in the planet/Sun mass ratio and perturbations from other planets; in particular, the absence of stable satellites around Saturn is mainly due to perturbations from Jupiter. It is, therefore, likely that satellites at distances 2r H could survive for the lifetime of the solar system around Uranus, Neptune, and, perhaps, Jupiter.