This paper deals with the theory of integration of scalar functions with respect to a measure with values in a, not necessarily locally convex, topological vector space. It focuses on the extension of such integrals from bounded measurable functions to the class of integrable functions, proving adequate convergence theorems, and establishing usable integrability criteria. The aim is to give an account that is both general, comprising all previously known formulations, and sufficiently simple, perhaps to be of use even to those who are interested only in the case of Banach space.