This book contains a systematic (and partly axiomatic) treatment of the holomorphic functional calculus for unbounded sectorial operators. The account is generic so that it can be used to construct and interrelate holomorphic functional calculi for other types of unbounded operators. Particularly, an elegant unified approach to holomorphic semigroups is obtained. The connection of functional calculus with interpolation theory is developed to a considerable extent. One chapter is dedicated to the stunning relations between functional calculus, similarity questions and numerical range conditions for operators on Hilbert spaces. In the last chapter applications to PDE, evolution equations and approximation theory as well as the connection with harmonic analysis are described.