Many of the phenomena that occur in the world around us are governed by nonlinear relationships. In the development of the mathematical sciences, the difficulties of nonlinear analysis have hindered the formulation of nonlinear concepts that would permit us to understand such phenomena. In the present article, our progress in understanding the behavior of nonlinear systems is reviewed and an attempt is made to present the resulting concepts in such a way that the} may be applied with some generality to other problems.