Cellular materials are widespread. Some, like wood and bone, occur in nature, while others, like polymeric foams, are manmade. Because of their cellular structure, they have unusual mechanical properties: they can be stiff, yet light, and they are capable of absorbing large deflections and thus large amounts of energy. Yet their mechanical behaviour has hardly been studied: no comprehensive attempt to relate mechanical properties to structure exists. In this thesis, we have attempted to do this. We first model a cellular material as a simple, two-dimensional array of hexagonal cells and identify and analyze the mechanisms by which it deforms. From this we calculate the elastic moduli and the elastic and plastic collapse stresses for ideal two-dimensional cellular materials. The results (which we have experimentally verified) show that each of these properties depends on three parameters: a solid cell wall material property, a geometric constant, and the relative density of the cellular material raised to the power two or three. We then examine three-dimensional cellular materials. Because their geometry is irregular and very complicated, no exact analysis of their behaviour is possible. But, with our understanding of two dimensional cellular materials and how they deform, we can use dimensional arguments to analyze three-dimensional cellular materials. The results of this analysis agree well with experimental data. Finally, we have applied our understanding of cellular materials to two case studies. In the first, we have examined the structure of cork, a quasi- two-dimensional cellular material, and explained some of its mechanical properties. The second case study analyzes the problem of material selection in packaging.