  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, we obtain an explicit characterization of all McVittie
spacetimes for which such orbits exist: there is a 2-parameter class of such
spacetimes, and so the existence of a circular photon orbit is a highly
specialised 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.
