This thesis presents an extensive computational study of electron-impact scattering and ionisation of atomic hydrogen and hydrogenic ions, which are fundamental to many diverse disciplines, from astrophysics and nuclear fusion to atmospheric physics. The non-relativistic Schrodinger equation describes these collisions, though finding solutions for even hydrogen, the simplest electron-atom collision, has proven to be a monumental task. Recently, Rescigno et al [Science 286, 2474 (1999)] solved this equation in coordinate space using exterior complex scaling (ECS), and presented the first electron-hydrogen differential cross sections for ionisation that matched with experiment without requiring uncontrolled approximation. This method has significant potential for extension to larger collision systems, but its large computational demand has limited its energy range and target configurations, and its application to discrete final-state collisions has been largely unexplored. Using radically different numerical algorithms, this thesis develops methods that improve the computational efficiency of ECS by two orders of magnitude. It extends the method to calculate discrete final-state scattering cross sections and enhances the target description to include hydrogenic ions and excited initial states. In combination, these developments allow accurate solutions over a broad range of energies and targets, for both scattering and ionisation, including the important near-threshold energy region where accurate calculations have been unavailable. The refined ECS method implemented in this work now offers complete numerical solutions of electron-hydrogen collisions, and its computational efficiency will facilitate its future application to more complex targets. The thesis culminates with the first ab initio quantum mechanical confirmation of ionisation threshold laws for electron-hydrogen collisions [Bartlett and Stelbovics, 2004, Phys. Rev. Lett. 93, 233201], which have resisted confirmation through the complete solution of the Schrodinger equation for more than half a century.