  We review recent progress in the description of the formation and evolution
of galaxy clusters in a cosmological context by using numerical simulations. We
focus our presentation on the comparison between simulated and observed X-ray
properties, while we will also discuss numerical predictions on properties of
the galaxy population in clusters. Many of the salient observed properties of
clusters, such as X-ray scaling relations, radial profiles of entropy and
density of the intracluster gas, and radial distribution of galaxies are
reproduced quite well. In particular, the outer regions of cluster at radii
beyond about 10 per cent of the virial radius are quite regular and exhibit
scaling with mass remarkably close to that expected in the simplest case in
which only the action of gravity determines the evolution of the intra-cluster
gas. However, simulations generally fail at reproducing the observed cool-core
structure of clusters: simulated clusters generally exhibit a significant
excess of gas cooling in their central regions, which causes an overestimate of
the star formation and incorrect temperature and entropy profiles. The total
baryon fraction in clusters is below the mean universal value, by an amount
which depends on the cluster-centric distance and the physics included in the
simulations, with interesting tensions between observed stellar and gas
fractions in clusters and predictions of simulations. Besides their important
implications for the cosmological application of clusters, these puzzles also
point towards the important role played by additional physical processes,
beyond those already included in the simulations. We review the role played by
these processes, along with the difficulty for their implementation, and
discuss the outlook for the future progress in numerical modeling of clusters.
