The variation of the Fermi energy with temperature and free-particle concentration is calculated analytically for a number of two-dimensional systems, including the free-particle gas, the intrinsic semiconductor and the impurity semiconductor. These illustrations, while, of course, applicable to certain physically realizable systems, are also of considerable illustrative and pedagogical interest, since the integrals which are involved can invariably be worked out in closed form. The Fermi energy as a function of temperature and concentration may then in some cases be calculated in closed form, while in others it is obtained implicitly as the solution to a fairly simple transcendental equation. The physical significance of the variation of Fermi energy as a function of temperature and concentration is discussed, and the deviation of then0p0product for the two-dimensional semiconductor from the Boltzmann mass-action result is calculated and discussed in detail. The conditions corresponding to the failure of the Boltzmann approximation and the onset of degeneracy are outlined. Application of the calculational methods to simple Bose-Einstein systems is briefly treated.