A stochastic model is presented to describe the structure and pore geometric properties of certain cellular micro and ultra filtration membranes. The model is based on a Voronoi tessellation, this is a contiguous space-exhaustive division of three-dimensional (3D) space. It is developed from a Poisson distribution of points each of which represents the nucleus of a cell which when developed coincides with an individual pore within the membrane space. The model is first applied to statistically homogeneous isotropic media and the statistics of the tessellation are computed and compared with the limited published data available. By applying first a constraint function and then a linear affine transformation to the tessellation a model is produced to simulate the structure of an anisotropic microfiltration membrane, cellulose acetate. The results computed from the model are compared with data obtained from image analysis of scanning election micrographs, SEM, of an actual membrane. Nonlinear transformations of the basic tessellation are applied to produce simulations of asymmetric cellular structures. The computed results are again compared with SEM image analysis data of polyether sulphone membranes. By applying a range of transformations the basic model can be used to simulate the structure of cellular homogeneous isotropic, anisotropic and asymmetric membranes. Furthermore, by using a combination of these the structure of composite membranes can also be modelled. Being based on statistical geometry the pore structure, shape, volume, cross-section area and perimeters are described by distribution functions and are not simply based on a mean length dimension implicit in all deterministic models which are used extensively in membrane research and application work. The implications to problems such as predicting the flux - transmembrane pressure characteristics, flux decline during filtration and backflashing are discussed.