Quantum mechanics provides a conventional theory of scattering that is limited in at least two ways: it focuses exclusively on the asymptotic regime and, in a more general sense characteristic of all quantum descriptions, provides no concrete account of individual particle evolution in spacetime. This is particularly true during the interaction. Along with a review of the conventional theory -- both formal and interpretive -- we introduce tools, such as Bohm's trajectory approach, that allow us to extend and sharpen our description of scattering events. Once the tools are developed and proven using a very simple toy model we apply them to the more realistic case of the Yukawa potential. The corresponding non-asymptotic probability densities, trajectories, differential cross sections, etc. are all considered. Bracketing the work we have also included a contextual discussion of the qualitative issues -- such as concerning the role of theory, interpretation, and pedagogy -- that necessarily accompany any theoretical extension of conventional topics.