Finding the exact solution of boundary and eigenvalue problems in the case where the boundary configuration is not natural to one of the common co-ordinate systems for which the governing partial differential equation can be solved by the standard method of separation of variables is, in general, out of the question. Approximate methods such as finite differences, finite elements, collocation, etc. must be used. It has been shown by several authors that the conformal mapping technique provides a suitable approach for analysing bending, buckling and vibration problems of thin elastic plates.