Several continuum mechanics models of the adhesion between elastic spheres have found application to compliant materials such as rubber and to fine particles in the air or in colloidal suspension. More recently they are being used in connection with experimental techniques such as the surface force apparatus and the atomic force microscope. The appropriate model to use depends on the conditions: the size and elasticity of the spheres and the load to which they are subjected. To guide this choice a map has been constructed with nondimensional coordinates &mgr; and &Pmacr;, where the elasticity parameter &mgr; can be interpreted as the ratio of elastic deformation resulting from adhesion to the effective range of surface forces and the load parameter &Pmacr; is the ratio of the applied load to the adhesive force. A closed form solution to the problem based on a Dugdale force-separation law is outlined and used to construct the map. The errors introduced by the Dugdale approximation are assessed by comparison with numerical solutions using the Lennard-Jones force law. Copyright 1997Academic Press