Abstract In this paper it is shown how the superposition method is employed to obtain accurate solutions for the free vibration frequencies and mode shapes of rectangular plates resting on elastic edge supports. The analysis is completely general in that uniform elastic rotational and translational supports of any stiffness magnitudes are considered to act simultaneously along each of the four edges, eight stiffness coefficients being required to define the problem. It is shown how all of the classical boundary conditions are approached when the coefficients are allowed to take on appropriate limits. A limited number of plots of fundamental mode eigenvalues vs. stiffness coefficients are presented for square plate problems of special interest. To the author's knowledge, this represents the first accurate comprehensive treatment of this problem to appear in the literature.