  We find the exact matrix model description of two dimensional Yang-Mills
theories on a cylinder or on a torus and with an arbitrary compact gauge group.
This matrix model is the singlet sector of a $c =1$ matrix model where the
matrix field is in the fundamental representation of the gauge group. We also
prove that the basic constituents of the theory are Sutherland fermions in the
zero coupling limit, and this leads to an interesting connection between two
dimensional gauge theories and one dimensional integrable systems. In
particular we derive for all the classical groups the exact grand canonical
partition function of the free fermion system corresponding to a two
dimensional gauge theory on a torus.
