This is the second of two volumes written to introduce the reader to some of the theories and methods which enable us to penetrate carefully and timorously into the nonlinear domain. Fortunately, we have no choice: the only direction is forward. In Volume I we focused on the basic concepts of stability and variational analysis and the analytic techniques required for their use such as linear differential equations and matrix analysis. In this volume we wish to describe some newer approaches such as duality techniques, differential inequalities, quasilinearization, dynamic programming, and invariant imbedding as well as some older methods which have become operational, and thus of greater analytic interest, as a consequence of the development of the digital computer: iteration, infinite systems of differential equations, and differential and integral quadrature. Using these theories we shall be able to present a more satisfactory explanation of various phenomena noted in Volume I, and indicate some promising new techniques.