Spectral Theory on the S Spectrum for Quaternionic Operators Introduction Slice hyperholomorphic functions The S spectrum and the S functional calculus Properties of the S functional calculus for bounded operators The S functional calculus for unbounded operators The H1 functional calculus The F functional calculus for bounded operators The F functional calculus for unbounded operators Quaternionic operators on a Hilbert space Spectral integrals The spectral theorem for bounded normal operators The spectral theorem for unbounded normal operators Spectral theorem for unitary operators Spectral Integration in the Quaternionic Setting Bounded Quaternionic Spectral Operators The subject of this monograph is the quaternionic spectral theory based on the notion of S spectrum. With the purpose of giving a systematic and self contained treatment of this theory that has been developed in the last decade the book features topics like the S functional calculus the F functional calculus the quaternionic spectral theorem spectral integration and spectral operators in the quaternionic setting. These topics are based on the notion of S spectrum of a quaternionic linear operator. Further developments of this theory lead to applications in fractional diffusion and evolution problems that will be covered in a separate monograph