  In any medium there are fluctuations due to temperature or due to the quantum
nature of its constituents. If a material body is immersed into such a medium,
its shape and the properties of its constituents modify the properties of the
surrounding medium and its fluctuations. If in the same medium there is a
second body then -- in addition to all direct interactions between them -- the
modifications due to the first body influence the modifications due to the
second body. This mutual influence results in a force between these bodies. If
the excitations of the medium, which mediate the effective interaction between
the bodies, are massless, this force is long-ranged and nowadays known as a
Casimir force. If the fluctuating medium consists of the confined
electromagnetic field in vacuum, one speaks of the quantum mechanical Casimir
effect. In the case that the order parameter of material fields fluctuates -
such as differences of number densities or concentrations - and that the
corresponding fluctuations of the order parameter are long-ranged, one speaks
of the critical Casimir effect. This holds, e.g., in the case of systems which
undergo a second-order phase transition and which are thermodynamically located
near the corresponding critical point, or for systems with a continuous
symmetry exhibiting Goldstone mode excitations. Here we review the currently
available exact results concerning the critical Casimir effect in systems
encompassing the one-dimensional Ising, XY, and Heisenberg models, the
two-dimensional Ising model, the Gaussian and the spherical models, as well as
the mean field results for the Ising and the XY model. Special attention is
paid to the influence of the boundary conditions on the behavior of the Casimir
force.
