Integrability of the AdS5 x S5 superstring and its deformations

This article reviews the application of integrability to the spectral problem of strings on AdS(5) x S-5 and its deformations. We begin with a pedagogical introduction to integrable field theories culminating in the description of their finite-volume spectra through the thermodynamic Bethe ansatz (TBA). Next, we apply these ideas to the AdS(5) x S-5 string and in later sections discuss how to account for particular integrable deformations. Through the AdS/CFT correspondence this gives an exact description of anomalous scaling dimensions of single trace operators in planar N = 4 supersymmetry Yang-Mills theory, its 'orbifolds', and beta and gamma-deformed supersymmetric Yang-Mills theory. We also 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 TBA description. Finally we discuss the TBA for a quantum deformation of the AdS(5) x S-5 superstring S-matrix, with close relations to among others Pohlmeyer reduced string theory, and briefly indicate more recent developments in this area.