This paper surveys models and statistical properties of random systems of hard particles. Such systems appear frequently in materials science, biology and elsewhere. In mathematical - statistical investigations, simulations of such structures play an important role. In these simulations various methods and models are applied, namely the RSA model, sedimentation and collective rearrangement algorithms, molecular dynamics, and Monte Carlo methods such as the Metropolis - Hastings algorithm. The statistical description of real and simulated particle systems uses ideas of the mathematical theories of random sets and point processes. This leads to characteristics such as volume fraction or porosity, covariance,contact distribution functions, specific connectivity number from the random set approach and intensity, pair correlation function and mark correlation functions from the point process approach. Some of them can be determined stereologically using planar sections, while others can only be obtained using three - dimensional data and 3D image analysis. They are valuable tools for fitting models to empirical data and, consequently, for understanding various materials, biological structures, porous media and other practically important spatial structures.