We investigate the equilibrium and stability of rigidly rotating quasi-neutral magnetospheres of objects with an aligned magnetic dipole moment, e.g., planetary magnetospheres. We use the Vlasov theory to calculate the equilibria with emphasis on trapped particle populations. Based on a new approach for the description of the trapped particles within the framework of collisionless theory, we calculate self-consistent axisymmetric magnetospheric models. For the purpose of demonstrating the capability of this approach, we choose a mathematically simple, but nevertheless physically relevant, class of distribution functions. By varying the plasma density, we calculate solution branches and investigate the bifurcation properties for typical cases. Furthermore, the stability of the solutions is tested by applying a sufficient stability criterion to them. We find that in our models there is always a maximum amount of plasma which can be confined in a rigidly rotating magnetosphere within a dipolelike magnetic field. The implications for possible scenarios of magnetospheric activity are discussed on the basis of these results. ¿ American Geophysical Union 1993