Formation of galaxy clusters corresponds to the collapse of the largest gravitationally bound overdensities in the initial density field and is accompanied by the most energetic phenomena since the Big Bang and by the complex interplay between gravity-induced dynamics of collapse and baryonic processes associated with galaxy formation. Galaxy clusters are, thus, at the cross-roads of cosmology and astrophysics and are unique laboratories for testing models of gravitational structure formation, galaxy evolution, thermodynamics of the intergalactic medium, and plasma physics. At the same time, their large masses make them a useful probe of growth of structure over cosmological time, thus providing cosmological constraints that are complementary to other probes. In this review, we describe our current understanding of cluster formation: from the general picture of collapse from initial density fluctuations in an expanding Universe to detailed simulations of cluster formation including the effects of galaxy formation. We outline both the areas in which highly accurate predictions of theoretical models can be obtained and areas where predictions are uncertain due to uncertain physics of galaxy formation and feedback. The former includes the description of the structural properties of the dark matter halos hosting clusters, their mass function, and clustering properties. Their study provides a foundation for cosmological applications of clusters and for testing the fundamental assumptions of the standard model of structure formation. The latter includes the description of the total gas and stellar fractions and the thermodynamical and nonthermal processes in the intracluster plasma. Their study serves as a testing ground for galaxy formation models and plasma physics. In this context, we identify a suitable radial range where the observed thermal properties of the intracluster plasma exhibit the most regular behavior and, thus, can be used to define robust observational proxies for the total cluster mass. Finally, we discuss the formation of clusters in nonstandard cosmological models, such as non-Gaussian models for the initial density field and models with modified gravity, along with prospects for testing these alternative scenarios with large cluster surveys in the near future.
