This book gives a somewhat unconventional introduction to  stochastic analysis.  Although most of the material covered here has appeared in other places, this book attempts to explain the core ideas on which that material is based.  As a consequence, the presentation is more an extended mathematical essay than a ``definition, lemma, theorem text.  In addition, it includes several topics that are not usually treated elsewhere.  For example, Wieners theory of homogeneous chaos is discussed, Stratovich integration is given a novel development and applied to derive Wong and Zakais approximation theorem, and examples are given of the application of Malliavins calculus to partial differential equations.  Each chapter concludes with several exercises, some of which are quite challenging.  The book is intended for use by advanced graduate students and research mathematicians who may be familiar with many of the topics but want to broaden their understanding of them.