These notes address the connection between two subjects, and they are thus intended to form an introduction to both but to be about neither. The discoveries of Fefferman and Stein about HP and BMO have interacted fruitfully with a great deal of work on the analogous ideas in martingale theory; the main goal of the following pages is an explanation of the fundamental result of Burkholder, Gundy, and Silver stein, which forms the bridge between these two areas of investigation. The exposition is at as elementary a level as possible, and it is intended in particular to be accessible to graduate students with a basic knowledge of measure theory, complex analysis and functional analysis. For the sake of those not familiar with probability theory, many probabilistic results are introduced and proved as needed, and there is a chapter without proofs on Brownian motion. Again, for those not on everyday terms with classical function theory, a survey of results on the maximal, square, and Littlewood-Paley functions is included, and function-theoretic arguments are given and estimates made in considerable detail. The discussion is restricted mainly to the case of the unit disk in the complex plane. I hope that one who reads these notes will find that GarsiaTs book, the papers of Fefferman and Stein, and the writings of Burkholder, Davis, Gundy, Herz, Silverstein, et al. on these topics are easily approachable.