Traditionally, convergence theorems for the stochastic approximation (SA) algorithm were along the lines of "IF the iterations are almost surely bounded, then the iterations converge to a solution" (subject to other conditions of course). A breakthrough paper by Borkar and Meyn (2000) made the almost sure boundedness of the iterates a part of the conclusions, rather than a part of the hypotheses. Their approach was based on the so-called ODE method, which shows that the sample paths of the SA algorithm converge to the deterministic solution trajectories of an ODE. In this talk, I will use an entirely different approach based on martingale convergence theory, originally proposed by Gladyshev (1965) for a limited class of problems. I will present a new converse Lyapunov theorem for globally exponentially stable ODEs. Using this, I am able to prove the convergence of the SA algorithm in a very streamlined fashion, and essentially reproduce the results of Borkar-Meyn. The new approach can, in principle, be extended to variations such as two-time scale (actor-critic) and function approximation versions of SA. That research is in progress.