© 2014 IOP Publishing Ltd. McVittie spacetimes represent an embedding of the Schwarzschild field in isotropic cosmological backgrounds. Depending on the scale factor of the background, the resulting spacetime may contain black and white hole horizons, as well as other interesting boundary features. In order to further clarify the nature of these spacetimes, we address this question: do there exist bound particle and photon orbits in McVittie spacetimes? Considering first circular photon orbits (CPOs), we obtain an explicit characterization of all McVittie spacetimes for which such orbits exist: there is a two-parameter class of such spacetimes, and so the existence of a CPO is a highly specialized feature of a McVittie spacetime. However, we prove that in two large classes of McVittie spacetimes, there are bound particle and photon orbits: future-complete non-radial timelike and null geodesics along which the areal radius r has a finite upper bound. These geodesics are asymptotic at large times to circular orbits of a corresponding Schwarzschild or Schwarzschild-de Sitter spacetime. The existence of these geodesics lays the foundations for and shows the theoretical possibility of the formation of accretion disks in McVittie spacetimes. We also summarize and extend some previous results on the global structure of McVittie spacetimes. The results on bound orbits are established using centre manifold and other techniques from the theory of dynamical systems.