This is a short survey of the historical development of the technique which has made possible an exact solution of a number of quantum field theories in one dimension, such as the quantized nonlinear Schrödinger equation and sine-Gordon equation. The development of the technique is described starting from the earlier works which established the inverse scattering transform for the same (and many more) equations in their classical form through the latest developments which have lead to appearance of a new fundamental mathematical concept of quantum groups. The links with other techniques, such as the direct approach to the integrable quantum field models based on the celebrated Bethe ansatz technique, and a fundamental similarity to the exact solutions, obtained by means of the transfer-matrix method, of a number of two-dimensional models of physics are highlighted, too. The main idea underlying this survey is a strong and unique interaction and mutual influence between physics and mathematics at this new sophisticated level.