A new definition of integral is proposed, which is based on a direct generalization of the standard Darboux sums; the new integral reduces to the ordinary one whenever the latter exists, to the partie-finie integral introduced by Hadamard (and later defined, equivalently, by many other authors,e.g. through methods of analytic regularization) otherwise. Our integral, for which we deem appropriate to retain the name of finite-part integral, must be defined for each specific class of singular integrands to be dealt with; this is sufficient to cope with all physical situations of interest,e.g. motion in general relativity, or ultraviolet divergences in quantum field theories; general criteria for doing so are indicated. The aim of this work is twofold: to provide the basis for a generalized, « quasi-local » theory of integration, and at the same time to supply an effective tool for actual use on the computer. For the latter reason, only the Riemann integral is considered in our generalization, Lebesgue integration being unfit for computer use (although theoretically better suited for the application of our concepts). The presentation is oriented mainly toward application; some discourses are kept on purpose at an intuitive level, to avoid the burden of complete formal treatments, which will be reported elsewhere.