  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 cluster, 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, the thermodynamical and non-thermal 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 intra-cluster plasma exhibit the most
regular behavior and thus can be used to define robust observational proxies
for the total cluster mass. We put particular emphasis on examining assumptions
and limitations of the widely used self-similar model of clusters. Finally, we
discuss the formation of clusters in non-standard 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.
