Publisher Summary This chapter explains the interaction between the Banach spaces and martingales, and the light this interaction sheds on the other parts of analysis, including the theory of singular integrals. To study the singular integral operators in a fairly general Banach space setting, one must surmount the difficulties caused by the lack of classical orthogonality. The same can be said for martingales and some other mathematical creatures that do not always have an easy transition from their original and friendly environment to one not quite so friendly. Certain kinds of biconvex functions designed to surmount these difficulties in the study of martingale transforms are just the fight tools to use in proving some of the most important sharp inequalities in the scalar-valued case. The ζ-convexity is defined and it is shown that a Banach space is unconditional for martingale differences (UMD) if it is ζ -convex. In addition to the biconvex function ζ, other biconvex and biconcave functions are also discussed in the chapter.