Andreas Kull IfA X-ray Emission of Clusters of Galaxies The X-ray emission of clusters of galaxies gives insight into the structure of clusters as well as it traces the peaks of the large scale distribution of matter in the Universe. In the first part of my talk I will discuss the statistical mechanical description of collisionless relaxation processes and its application to mixed dark matter models and clusters of galaxies. A simple self-consistent model of clusters of galaxies is presented and its predicted X-ray emission is compared to ROSAT observations. Preliminary results from the study of ROSAT All-Sky Survey data covering the Shapley region will be discussed in the second part of the talk. The Shapley supercluster is one of the densest large scale concentrations of matter in the local Universe ($z<0.1$). I will present evidence for elongated, faint X-ray emission in the vicinity of the clusters of galaxies A3562, A3558 and A3556 spanning $\sim 17 h_{50}^{-1}$ Mpc.