The homogeneous spatial domains of phases on a mesoscopic scale are a characteristic feature of many composite media such as complex fluids or porous materials. The thermodynamics and bulk properties of such composite media depend often on the morphology of its constituents, i.e., on the spatial structure of the homogeneous domains. Therefore, a statistical theory should include morphological descriptors to characterize the size, shape and connectivity of the aggregating mesophases. We propose a new model for studying composite media using morphological measures to describe the homogeneous spatial domains of the constituents. Under rather natural assumptions a general expression for the Hamiltonian can be given by extending the model of Widom and Rowlinson [B. Widom, J.S. Rowlinson, J. Chem. Phys. 52 (1970) 1670-1684] for penetrable spheres. The Hamiltonian includes energy contributions related to the volume, surface area, mean curvature, and Euler characteristic of the configuration generated by overlapping sets of arbitrary shapes. A general expression for the free energy of composite media is derived and we find that the Euler characteristic stabilizes a highly connected bicontinuous structure resembling the middle-phase in oil-water microemulsions for instance. (C) 1998 Elsevier Science B.V. All rights reserved.