In the first section of this chapter we give a brief survey of the theory of slice hyperholomorphic functions, and we provide the basic results needed in the sequel. The theory is much more developed and we refer the reader to the Introduction to Part II of the present work for references. Then, the Sections from 6.2 to 6.5 focus on the functions slice hyperholomorphic in the unit ball. We define in particular the Hardy space of the ball, Schur functions, and Blaschke factors. We also study linear fractional transformations and introduce the Wiener algebra of the ball. The last two sections, 6.6 and 6.7, are devoted to the case of functions slice hyperholomorphic in the half-space of quaternions with real positive part. We discuss in particular the Hardy space and Blaschke factors.