Scientists and engineers are often confronted with a non-linear ordinary differential equation for which physical considerations suggest that one or more types of oscillatory motions are possible. In general, except under very special conditions, exact solutions in closed form do not exist. Therefore, if analytic results on the differential equation are to be obtained, then approximation schemes must be constructed. For the case where the non-linearities are small a number of perturbation methods can be applied and approximate analytic solutions obtained. However, one is often interested in looking at situations where the system is operating in a regime where the non-linear interactions are very large; consequently, perturbation techniques will not work. To my knowledge, the only technique of general applicability for obtaining approximate analytic solutions under these conditions is the method of harmonic balance. In general, when the method works it provides an excellent approximation to the required solution.