Inverse quantum scattering theory has proved to be one of the mast fruitful discoveries in physics. It provides model-independent microscopical interactions in quantum physics and supplies fundamental theoretical tools to solve problems in nonlinear physics. Founded in the 1950s, inverse quantum scattering theory has originated from the spectral theory of the Schrodinger equation or, more precisely, the Sturm-Liouville operator. It soon became the heart of the inverse scattering transform, enabling tremendous technical applications such as wave propagation, light transmission through optical fibres, and so on. However, its original goal to derive quantum potentials from observables has been reached only in recent decades by suitable modifications of the theory. Also, other quantum inversion schemes have been developed. One of the purposes of this volume is to give an account of this progress.