We have generalized the Maugis model of the adhesive contact of spheres (Maugis D., J. Colloid Interface Sci. 150, 243, 1992) to arbitrary interactions in two different ways. On the one hand, if we choose an arbitrary interaction potential between surfaces, we obtain an exact formulation of the problem, which unfortunately requires computationally demanding numerical solutions, On the other hand, if we prescribe the stress distribution outside the contact zone, we can build numerically efficient approximations which completely describe the contact problem for a given monotonic attractive interaction with reasonable accuracy. These approximate models retain the simplicity of the Maugis model and may therefore prove useful in data analysis. Our results show that the nature of the interaction has a very moderate influence on the solution, so that the Maugis model can be used to describe the contact of spheres with most physically acceptable interaction potentials, except when large accuracy is required.