Spinning Strings and Correlation Functions in the AdS CFT Correspondence Part I: Introduction Integrability in the AdS CFT Correspondence Part II: Integrability on the String Theory Side Strings in Coset Spaces Flux deformed Neumann Rosochatius System η Deformed Neumann Rosochatius System Part III: Integrability on the Field Theory Side. Spin Chains Introduction: The Two Bethe Ansätze Two points Functions and ABA Tailoring and Hexagon Form Factors Part IV: Conclusions Summary and Conclusions This book addresses several aspects of the integrable structure of the AdS CFT correspondence. In particular it presents computations made on both sides of the AdS CFT correspondence at weak and at strong coupling. On the string theory side of the correspondence the book focuses on the evaluation of the energy spectrum of closed string solutions moving in some deformed backgrounds that preserve integrability. On the gauge theory side it explores various formal problems arising in the computation of two and three point functions by means of the Algebraic Bethe Ansatz and the Quantum Inverse Scattering method. The book features numerous results on integrability in the context of the AdS CFT correspondence. Self contained and pedagogical it includes general discussions and detailed presentations on the use of integrable systems techniques and their applications