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The one-dimensional Heisenberg XXX spin chain appears in a special limit ofthe AdS/CFT integrable system. We review various ways of proving itsintegrability, and discuss the associated methods of solution. In particular,we outline the coordinate and the algebraic Bethe ansatz, giving reference toliterature suitable for learning these techniques. Finally we speculate whichof the methods might lift to the exact solution of the AdS/CFT system, andsketch a promising method for constructing the Baxter Q-operator of the XXXchain. It allows to find the spectrum of the model using certain algebraictechniques, while entirely avoiding Bethe's ansatz.

Review of AdS/CFT Integrability, Chapter III.1: Bethe Ans\"atze and the R-Matrix Formalism