This paper describes a serendipitous galaxy cluster survey that we plan to conduct with the XMM X-ray satellite. We have modeled the expected properties of such a survey for three different cosmological models, using an extended Press-Schechter formalism combined with a detailed characterization of the expected capabilities of the European Photon Imaging Camera (EPIC) camera on board XMM. We estimate that, over the 10 yr design lifetime of XMM, the EPIC camera will image a total of≃ 800 deg 2 in fields suitable for the serendipitous detection of clusters of galaxies. For the presently favored low-density model with a cosmological constant, our simulations predict that this survey area would yield a catalog of more than 8000 clusters, ranging from poor to very rich systems, with around 750 detections above z= 1. A low-density open universe yields similar numbers, though with a different redshift distribution, while a critical-density universe gives considerably fewer clusters. This dependence of catalog properties on cosmology means that the proposed survey will place strong constraints on the values of Ω 0 and Ω Λ. The survey would also facilitate a variety of follow-up projects, including the quantification of evolution in the cluster X-ray luminosity-temperature relation, the study of high-redshift galaxies via gravitational lensing, follow-up observations of the Sunyaev-Zeldovich effect, and foreground analyses of cosmic microwave background maps.