The primary aim of this chapter is to develop a general theory of integration, that is, a theory of integration for real valued functions defined on abstract spaces with respect to abstract measures. In particular, when applied with Lebesgue measure on the real line, this new integration theory will take us way beyond Riemann integration theory. It will enable us to define integrals for a much larger class of functions than the ones for which Riemann theory works, while, at the same time, the new integral will agree with the classical Riemann integral when the latter exists.