This article deals with vector integration and stochastic integration in Banach spaces. In particular, it considers the theory of integration with respect to vector measures with finite semivariation and its applications. This theory reduces to integration with respect to vector measures with finite variation which, in turn, reduces to the Bochner integral with respect to a positive measure. The article describes the four stages in the development of integration theory. It first provides an overview of the relevant notation for Banach spaces, measurable functions, the integral of step functions, and measurability with respect to a positive measure before discussing the Bochner integral. It then examines integration with respect to measures with finite variation, semivariation of vector measures, integration with respect to a measure with finite semivariation, and stochastic integrals. It also reviews processes with integrable variation or integrable semivariation and concludes with an analysis of martingales.