This second edition preserves the original text of 1968, with clarification and added references. From the Preface to the Second Edition: ""Since the First Edition of this book, numerous important results have appeared--in particular stochastic integrals with respect to martingales, random fields, Riccati equation theory and realization of nonlinear filters, to name a few. In Appendix D, an attempt is made to provide some of the references that the authors have found useful and to comment on the relation of the cited references to the field ...[W]e hope that this new edition will have the effect of hastening the day when the nonlinear filter will enjoy the same popularity in applications as the linear filter does now.