Method of Darboux Transformations in the Theory of Integrable Systems: 30 years of development

We explain how the dispersed observations of Darboux, Moutard, Frobenius, and Crum, (essentially coming back to the 19-th century), concerning the linear ODE and mainly Sturm-Liouville equations, were unified and generalized to the hierarchies of linear and nonlinear PDE, providing one of the most powerfull tools in the theory of integrable systems and the theory of orthogonal polynomials. We start from explanation of the historical background material and next present some of the main developments of the last 30 years.
The content of this lecture can be easily understood by everybody familiar with linear algebra and elementary analysis: it will be adressed to the large audience with no a prior knowledge of the theory of integrable systems.