  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. Next, we
apply these ideas to the AdS_5 x S^5 string and in later chapters 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 thermodynamic Bethe ansatz description. Finally we
discuss the thermodynamic Bethe ansatz 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.
