In the integrals of Riemann, Newton, Lebesgue, Perron, Kurzweil a given function is integrated with respect to the identity function. Some physical problems have exacted the extension of concept of integral to integrals in which the given function is integrated with respect to a function which does not have to be the identity in general. The first time when such integral appeared was in a famous Stieltjes' treatise dedicated to the connection between the convergence of chain fractions and the problem how to describe the distribution of matter on a solid line segment when all moments of this line segment of natural orders are known. The integrals of this kind have been since then called Stieltjes integrals. To various modifications of the definition, which with time arose, the names of the authors of these modifications are then usually added. Soon there were integrals of: Riemann-Stieltjes, Perron-Stieltjes or Lebesgue-Stieltjes, and the definition of the Stieltjes integral in Kurzweil's sense.
The important applications of the Kurzweil-Stieltjes integral are in functional analysis, in the theory of various kinds of generalized differential equations, including equations with impulses and dynamic equations on time scales, and in other fields such as stochastics process, hysteresis, and financial market.