This dissertation describes an investigation of systems of polydisperse smooth hard spheres. This includes the development of a fast contact detection algorithm for computer modelling, the development of macroscopic constitutive laws that are based on microscopic features such as the moments of the particle size distribution, and the development of new analysis techniques to study microstructure in such systems with different physical behaviour, i.e., gaseous, liquid, glassy and crystalline states. The first chapter gives a general introduction to the themes and topics treated in this dissertation. In the second chapter I deal with the numerical problem of contact detection among arbitrarily polydisperse objects. I present a new efficient algorithm for contact detection which even increases performance with increasing degree of polydispersity. The performance of the algorithm is theoretically analyzed for various particle size distributions and volume fractions, and recommendations are given concerning the choice of optimal algorithm parameters. The third chapter focuses on the theoretical prediction of the equation of state and the jamming density. The equilibrium equation of state of a fluid mixture of polydisperse hard spheres is well described by considering only the first three moments of the size distribution function. Consequently, the (thermodynamic) properties of a polydisperse fluid can be reproduced by a well-chosen “equivalent” bidisperse fluid with the same three moments. In this study I ask the question: How many moments are needed to predict the pressure and the jamming density of polydisperse mixtures in compressed non-equilibrium glassy states? I find that five moments suffice to describe the properties of polydisperse mixtures for all densities, including glassy, nonequilibrium states and the maximal jamming density. Hence, as proposition, polydisperse mixtures can be modelled by a well-chosen tridisperse system. In the fourth chapter I suggest a new way to characterize the microstructure in mono- and polydisperse hard-sphere systems, based on the local correlation of four particle positions. This analysis allows to distinguish between gaseous, liquid, partially and fully crystallized, and glassy (random) jammed states. A common microstructural feature is observed in crystalline and glassy jammed states, suggesting the presence of “hidden” two-dimensional order in polydisperse random close packings of three-dimensional spheres. Finally, conclusions and outlook close the dissertation.