The basic stochastic approximation algorithms introduced by Robbins and Monro and by Kiefer and Wolfowitz in the early 1950s have been the subject of an enormous literature, both theoretical and applied. This is due to the large number of applications and the interesting theoretical issues in the analysis of “dynamically defined” stochastic processes. The basic paradigm is a stochastic difference equation such as θn+1 = θn +εnYn, where θn takes its values in some Euclidean space, Yn is a random variable, and the “step size” εn > 0 is small and might go to zero as n → ∞. In its simplest form, θ is a parameter of a system, and the random vector Yn is a function of “noise-corrupted” observations taken on the system when the parameter is set to θn. One recursively adjusts the parameter so that some goal is met asymptotically. This book is concerned with the qualitative and asymptotic properties of such recursive algorithms in the diverse forms in which they arise in applications. There are analogous continuous time algorithms, but the conditions and proofs are generally very close to those for the discrete time case.