In the last chapter, we presented a general overview of classical integrable systems and their essential features. Although it has not been possible to construct a quantum mechanical counterpart of every classical technique, remarkable progress has been made in the developement of quantum integrable systems. Regarding quantization of nonlinear equations, the simplest approach is to consider the nonlinear ﬁeld occurring in a nonlinear equation as an operator and assume the equation to be derivable from a suitable Hamiltonian with a well-deﬁned commutation rule. Indeed this was the essential approach adopted by Bethe in his treatment of the many-body bosonic system with a δ-function potential . This system is today better known as the δ-function Bose gas. Later, it was observed that such a system is equivalent to the nonlinear Schro¨dinger equation. We start therefore with a very brief introduction to the Bose gas, adopting the approach as outlined in the celebrated article by Fowler .