**Abstract:**

AdS/CFT integrability amounts to the statement that part of the dynamics of AdS strings / corresponding fields can be obtained by solving a quantum integrable system. This has been turned into a powerful tool to obtain exact information from the two sides of the correspondence, and has revealed a large amount of symmetry hidden in the system. We will review the state of the art of this activity, with focus on the spectral problem.

The first part will present a short survey of the traditional stages of solution of a quantum integrable system using toy examples. At the end of each stage, the main challenges for the application to the AdS/CFT case will be pointed out, and referred to the second part for their description. The topics covered will be:

- Spin chains and Bethe ansatz
- S-matrix and its analytic structure
- Lax pairs and quantum inverse scattering method
- Hopf algebras and Yangians
- (depending on time) Thermodynamic Bethe ansatz

The second part will analyse in more detail how the challenges hinted at in the first part are tackled, with two main examples in mind: strings in five-dimensional AdS (dual to N=4 super Yang Mills fields) and strings in three-dimensional AdS, dual to two-dimensional conformal field theories. The description will cover:

- Strings in lightcone gauge and coset sigma-models
- Worldsheet QFT and its symmetries
- The non-relativistic S-matrix and relation to spin-chains
- Finding the spectrum

For each point we will emphasise the state of the art and point out the current directions of research.

**References:**

Inverse scattering method and S-matrix:

- O. Babelon, D. Bernard, M. Talon, "Introduction to Classical Integrable Systems", Cambridge U. Press, 2003.
- Faddeev, L. D. "How Algebraic Bethe Ansatz works for integrable model." arXiv preprint hep-th/9605187 (1996).
- Dorey, Patrick. "Exact S-matrices." Conformal field theories and integrable models. Springer Berlin Heidelberg, 1997. 85-125.
- Sklyanin, E. K. "Quantum version of the method of inverse scattering problem." Journal of Soviet Mathematics 19.5 (1982): 1546-1596.
- Thacker, H. B. "Exact integrability in quantum field theory and statistical systems." Reviews of Modern Physics 53.2 (1981): 253.

Hopf algebras and Quantum groups:

- Jimbo, M. "Topics from representations of Uq(g). An introductory guide to physicists." Nankai Lectures on Mathematical Physics (1992): 1-61.
- Kassel, Christian. "Quantum groups". Volume 155 of Graduate Texts in Mathematics. Berlin, Springer Verlag (1995).
- Delius, Gustav W. "Exact S-matrices with affine quantum group symmetry." Nuclear Physics B 451.1 (1995): 445-465.

AdS/CFT Integrability:

- Beisert, Niklas, et al. "Review of AdS/CFT integrability: an overview." Letters in Mathematical Physics 99.1-3 (2012): 3-32.
- Arutyunov, Gleb, and Sergey Frolov. "Foundations of the AdS5×S5 superstring: I." Journal of Physics A: Mathematical and Theoretical 42.25 (2009): 254003.
- A. Babichenko, B. Stefanski Jr., K. Zarembo "Integrability and the AdS(3)/CFT(2) correspondence", JHEP 1003 (2010)058, arXiv:0912.172.
- R. Borsato, O. Ohlsson Sax, A. Sfondrini, "A dynamic su(1|1)^2 S-matrix for AdS3/CFT2", JHEP 1304 (2013) 113, arXiv:1211.5119.
- R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefanski, Jr., A. Torrielli, "Dressing phases of AdS3/CFT2", Phys.Rev. D88 (2013) 066004, arXiv:1306.2512.