We consider the structural and physical properties of complex model morphologies and microstructures obtained by Xray-CT imaging. The Minkowski functionals, a family of statistical measures based on the Euler-Poincar&eacute characteristic of n-dimensional space, are shown to be sensitive measures of the morphology of disordered structures. Analytic results for the Boolean model are given and used to devise a reconstruction scheme, which allows one to accurately reconstruct a complex Boolean structure given at any phase fraction for all other phase fractions. The percolation thresholds of either phase are obtained with good accuracy. From the reconstructions one can subsequently predict property curves for the material across all phase fractions from a single 3D image. We illustrate this for transport and mechanical properties of complex Boolean systems and for experimental sandstone samples. By extending the Minkowski functionals to parallel surfaces using operations from mathematical morphology, a powerful discrimination of structure is obtained. Further the sensitivity of the Minkowski functionals under experimental conditions is analysed. Accurate calculations of conductive and elastic properties directly from tomographic images are achieved by estimating and minimising several sources of numerical error. Simulations of electrical conductivity and linear elastic properties on microtomographic images of Fontainebleau sandstone are in excellent agreement with experimental measurements over a wide range of porosity. The results show the feasibility of combining digitised images with transport and elasticity calculations to accurately predict physical properties of individual material morphologies. We show that measurements of properties based on microtomographic images are more accurate than those based on conventional theories for disordered materials. We study the elastic behaviour of model clean and cemented sandstones. Results are in excellent agreement with available experimental data, and are compared to conventional theoretical and empirical laws. A new predictive empirical method is given for predicting the elastic moduli of sandstone morphologies. The method gives an excellent match to numerical and experimental data.