Functions of a quaternionic variable with properties generalizing holomorphicity from the complex to the quaternionic setting can be defined in various different ways. The notion of hyperholomorphicity which looks more suitable for some applications to operator theory is the so-called slice hyperholomorphicity; this is the main object of this chapter. This function theory has been evolving rapidly and for more information, the reader may consult the books [23, 58, 60, 72, 76] and the references therein. This chapter also contains some basic notions on the so-called S-functional calculus which is nowadays very well developed. Dozens of papers have been written on this topic, in various directions, and here with no sake of completeness we mention [12–14, 16, 18, 22, 52, 54–57, 59, 74]. For more details, we refer to [23, 53, 60] and the references therein.