In classical analysis, the Riemann - Stieltjes integral was the first attempt to generalize the idea of the size, or measure, of a subset of the real numbers. Instead of simply using the length of an interval as a measure, in this chapter, the author uses any function that satisfies the same properties as the length function. He easily proves the usual properties that his expects an integration type mapping to have. The author turns our attention to the question of what pairs of functions might have a Riemann - Stieltjes integral. He learns how to deal with integrators that are monotone functions. The author extends the notion of Darboux Upper and Lower Sums in the obvious way. The connection between the Riemann - Stieltjes and Riemann - Stieltjes Darboux integrals is obtained using an analog of the familiar Riemann Condition.