A Modern View of the Riemann Integral This monograph uncovers the full capabilities of the Riemann integral. Setting aside all notions from Lebesgue’s theory the author embarks on an exploration rooted in Riemann’s original viewpoint. On this journey we encounter new results numerous historical vignettes and discover a particular handiness for computations and applications. This approach rests on three basic observations. First a Riemann integrability criterion in terms of oscillations which is a quantitative formulation of the fact that Riemann integrable functions are continuous a.e. with respect to the Lebesgue measure. Second the introduction of the concepts of admissible families of partitions and modified Riemann sums. Finally the fact that most numerical quadrature rules make use of carefully chosen Riemann sums which makes the Riemann integral be it proper or improper most appropriate for this endeavor. A Modern View of the Riemann Integral is intended for enthusiasts keen to explore the potential of Riemanns original notion of integral. The only formal prerequisite is a proof based familiarity with the Riemann integral though readers will also need to draw upon mathematical maturity and a scholarly outlook.