Integrability in gauge/string dualities

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Integrability in gauge/string dualities will be reviewed in a broad perspective with a particular emphasis on the recently proposed equations describing the full planar spectrum of anomalous dimensions in AdS/CFT [N.Gromov, V.Kazakov, PV]. These are a concise version of Thermodynamic Bethe equations, called Y-system, which generalize the asymptotic Bethe equations of Beisert and Staudacher (which yield the full spectrum of N=4 SYM for asymptotically long local operators) and incorporate the 4-loop results for the shortest twist two operators obtained by Bajnok and Janik from the dual string sigma model (thus reproducing perturbative gauge theory computations with thousands of diagrams). On the way, we will explain some of the interesting open problems in the field.