  I discuss many-body models for interacting fermions in two space dimensions
which can be solved exactly using group theory. The simplest example is a model
of a quantum Hall system: 2D fermions in a constant magnetic field and a
particular non-local 4-point interaction. It is exactly solvable due to a
dynamical symmetry corresponding to the Lie algebra $\gl_\infty\oplus
\gl_\infty$. There is an algorithm to construct all energy eigenvalues and
eigenfunctions of this model. The latter are, in general, many-body states with
spatial correlations. The model also has a non-trivial zero temperature phase
diagram. I point out that this QH model can be obtained from a more realistic
one using a truncation procedure generalizing a similar one leading to mean
field theory. Applying this truncation procedure to other 2D fermion models I
obtain various simplified models of increasing complexity which generalize mean
field theory by taking into account non-trivial correlations but nevertheless
are treatable by exact methods.
