**Abstract:** In 1966, D. L. Burkholder published his seminal paper on the boundedness of martingale transform. While the main results were of considerable importance, the ideas introduced in this paper became the building blocks for many subsequent results in martingales, leading to the celebrated Burkholder-Davis-Gundy inequalities (the bread and butter of modern stochastic analysis) and to numerous applications in analysis, including the solution by Burkholder, Gundy and Silverstein of a longstanding open problem concerning a real variables characterization of the Hardy spaces. In 1984, motivated in part by applications to the geometry of Banach spaces, Burkholder gave a new proof of his 1966 inequality which bypassed L^{2} theory, introducing a novel and revolutionary new method for proving optimal inequalities in probability and harmonic analysis. This method, commonly referred to now days as the "Burkholder method" (and also as "the Bellman function method"), has had many applications in recent years. The aim of these lectures is to present some of these martingale inequalities and applications, emphasizing ideas rather than technicalities, taking a historical point of view but discussing applications to problems of current interest.

Lecture 1: The 1966 martingale inequalities, the good-principle, some applications.

Lecture 2: Sharp inequalities. The "why" and the "how".

Lecture 3: Optimal bounds for singular integrals, Fourier multipliers, connections to rank-one convexity, quasi convexity and weighted norm inequalities.