This chapter presents the various microscopic methods used to tackle one-dimensional systems: the famous Bethe-ansatz and numerical techniques. In addition to the Bethe-ansatz, various numerical methods that have become incredibly accurate in one dimension are also examined, including exact diagonalizations. Without entering into the technical details, a short sketch of the various methods with their pros and cons is given. This chapter also shows how to combine these methods with the bosonization method to get the best of both worlds. Furthermore, the Bethe-ansatz solution is coupled with the Luttinger liquid concept to compute the Luttinger liquid parameters for the spin chain.
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