Because the response of a magnetic substance to an applied field depends strongly on the physical properties of the material, much can be learned by monitoring that response through what is known as a ''magnetic hysteresis loop.'' The measurements are rapid and quickly becoming part of the standard set of tools supporting paleomagnetic research. Yet the interpretation of hysteresis loops is not simple. It has become apparent that although classic ''single-domain,'' ''pseudo-single-domain'' and ''multidomain'' loops described in textbooks occur in natural samples, loops are frequently distorted, having constricted middles (wasp-waisted loops) or spreading middles and slouching shoulders (potbellies). Such complicated loops are often interpreted in oversimplified ways leading to erroneous conclusions. The physics of the problem have been understood for nearly half a century, yet numerical simulations appropriate to geological materials are almost unavailable. In this paper we discuss results of numerical simulations using the simplest of systems, the single-domain/superparamagnetic (SD/SP) system. Examination of the synthetic hysteresis loops leads to the following observations: (1) Wasp-waisting and potbellies can easily be generated from populations of SD and SP grains. (2) Wasp-waisting requires an SP contribution that saturates quickly resulting in a steep initial slope, and potbellies require low initial slopes (the SP contribution approaching saturation at higher fields). The approach to saturation is dependent on volume hence the cube of grain diameter. Therefore there is a very strong dependence of hysteresis loop shape on the assumed threshold size. (3) We were unable to generate potbellies using a SP/SD threshold size as large as 30 nm, and wasp waists cannot be generated using a threshold size as small as 8 nm. The occurrence of both potbellies and wasp waists in natural samples is consistent with a room temperature threshold size of some 15 nm (¿5). (4) Simulations using a threshold size of 15--20 nm with populations dominated by SP grain sizes, that is with a small number of SD grains, produce synthetic hysteresis loops consistent with measured hysteresis loops and transmission electron microscopic observations from submarine basaltic glass. (5) Simulations and measurements using two populations with distinct coercivity spectra can also generate wasp-waisted loops. A relatively straightforward analysis of the resulting loops can distinguish the latter case from wasp-waisting resulting from SP/SD behavior. ¿ American Geophysical Union 1996