The geometry of an "ideal" sponge is defined and described with reference to single-sheeted hyperbolic surfaces. The expected swelling features of this sponge, which depend on the detailed swelling mechanism, can be used to deduce estimates of the structural parameters of the sponge. The analysis is used to investigate the mesostructure of some sponge mesophases, in bulk and confined between mica sheets, relying on data collected by others. Sponge mesophases formed in sodium dodecyl sulphate-pentanol-NaCl-water (Europhys. Lett. 1989, 9, 447-452) and Aerosol OT-NaCl-water (Structure and dynamics of strongly interacting colloids and supramolecular aggregates in solution; Chen, S.-H., Ed.;Kluwer Academic Publishers : Dordrecht, The Netherlands, 1992, pp 351-363) mixtures are analyzed, as well as the sponge phases confined between mica sheets (in surface force experiments) in AOT-NaNO3-water mixtures (J. Phys. II 1995, 5, 103-112 and Langmuir 1995, 11,3928-3936). In addition, we analyze a cubic mesophase formed in the glycerol monooleate-water system (Nature 1994, 368, 224-226) and the AOT-NaCl-water lamellar mesophase (Structure and dynamics of strongly interacting colloids and supramolecular aggregates in solution; Chen, S.-H., Ed.; Kluwer Academic Publishers : Dordrecht, The Netherlands, 1992, pp 351-363). The analyses suggest that the sponge mesophases considered have a structure related to that of bicontinuous cubic mesophases, viz. a hyperbolically warped molecular bilayer, dividing two continuous, but disjointed, water labyrinths. The diffuse scattering peak, observed in small-angle scattering spectra of these sponges, corresponds to the average channel diameter in bulk systems but is more complex in the confined case, which appears to be superstructured. The swelling analysis is shown to be equally useful for detailed investigation of the effective (average) topology of other mesophases (the cubic and lamellar phases). The power of the analysis lies in its ability to detect the average topology of a membrane, provided the membrane is sufficiently ordered to furnish at least one correlation peak in small-angle scattering spectra.