The present work is a continuation of that in earlier volumes by Ames and Rogers and Shadwick. Here, the object is to give an up-to-date account of analytic methods available for the exact solution of nonlinear boundary value problems of physical interest. Thus, the text may be regarded as supplementing the numerous texts now in print on the application of the inverse scattering transform to the solution of nonlinear initial value problems. Here, the nonlinearities to be considered may reside in the equations to be solved or in the prevailing boundary conditions. In either case, only privileged classes of nonlinear problems are amenable to exact solutions.