The book contains a full exposition of a new functional calculus for n-tuples of not necessarily commuting linear operators and of its analog for quaternionic linear operators. These calculi are based on the recently developed theory of hyperholomorphicity for functions with values in a Clifford algebra (the so-called slice monogenic functions) which are carefully described in the book and, in particular, for functions with values in the algebra of quaternions, the so-called slice regular functions.