Martingale theory is a cornerstone of modern probability, offering a natural extension of the study of sums of independent random variables. Although its roots can be traced back to the work of Paul Lévy in 1937, it was Joseph L. Doob in the 1940s who formally developed the theory, culminating in his landmark book Stochastic Processes in 1953. Since then, martingale theory has evolved significantly, with deep contributions from mathematicians such as Donald L. Burkholder, Richard Gundy, and Burgess Davis, among others. This is what is now known as advanced martingale theory, which began with the publication of Burkholder's seminal paper Martingale Transforms in 1966.

This book provides a comprehensive treatment of both classical and advanced martingale theory. It opens with a historical introduction, exploring foundational functions such as Rademacher, Haar, and Walsh functions, before delving into the core concepts of conditional probability. The classical theory, as developed by Doob, is meticulously presented, followed by an in-depth examination of modern advancements, including Burkholder's inequalities, Burkholder-Davis-Gundy inequality, and their generalizations, as well as good-lambda inequalities. The final chapter showcases a wide range of applications, highlighting the theory's profound impact on Banach space theory, harmonic analysis, and beyond.

Intended for graduate students and researchers in probability and analysis, this book serves as both an introduction and a reference, offering a clear and structured approach to a subject that continues to shape mathematical research and its applications.

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Ricardo Rios is an Associate Professor at the Facultad de Ciencias, Universidad Central de Venezuela. He holds a Ph.D. in Mathematics from the University of Paris-Saclay (Paris XI) and an M.Sc. and B.Sc. in Mathematics from Universidad Central de Venezuela. His research focuses on nonparametric functional estimation with dependent data, probability theory, statistics, and stochastic processes, with applications in martingale theory and machine learning.

Wilfredo Urbina-Romero is an Associate Professor at Roosevelt University. He earned his Ph.D. in Mathematics from the University of Minnesota and his M.Sc. and B.Sc. in Mathematics from Universidad Central de Venezuela. His research interests include harmonic analysis, orthogonal polynomial theory, and martingale theory.