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This article reviews the application of integrability to the spectral problemof strings on AdS_5 x S^5 and its deformations. We begin with a pedagogicalintroduction to integrable field theories culminating in the description oftheir finite-volume spectra through the thermodynamic Bethe ansatz. Next, weapply these ideas to the AdS_5 x S^5 string and in later chapters discuss howto account for particular integrable deformations. Through the AdS/CFTcorrespondence this gives an exact description of anomalous scaling dimensionsof single trace operators in planar N=4 supersymmetry Yang-Mills theory, its`orbifolds', and beta and gamma-deformed supersymmetric Yang-Mills theory. Wealso touch upon some subtleties arising in these deformed theories.Furthermore, we consider complex excited states (bound states) in the su(2)sector and give their thermodynamic Bethe ansatz description. Finally wediscuss the thermodynamic Bethe ansatz for a quantum deformation of the AdS_5 xS^5 superstring S-matrix, with close relations to among others Pohlmeyerreduced string theory, and briefly indicate more recent developments in thisarea.

Integrability of the AdS_5 x S^5 superstring and its deformations