A minimum-norm method has been developed for solving the coupled integro-differential equations describing the scattering of positrons by one-electron targets in which the rearrangement channels for positronium formation have been explicitly included. The minimum-norm method, applied to this application for the first time in this thesis, is an enhancement of a previously reported least-squares method which has enabled the extension to a significantly larger basis consisting of up to 26 states on the direct centre, including pseudostates, and 3 states on the positronium. The method has been applied here to e+-H and e+-He+ scattering; cross-sections have been produced for the latter over a range of energies up to 250 eV. The basis was found to be large enough to produce smooth cross sections and little evidence of pseudoresonance structure was found. The results are the first converged cross sections to be calculated for e+-He+ scattering using the coupled channel approximation. Results for e+-H scattering compare well with the work of other authors. A highly efficient parallel code was developed for solving the largest coupling cases. The results prove the minimum-norm approach to be an accurate and reliable method for large-scale coupled channel calculations involving rearrangement collisions. Also in this thesis, the capture of slow antiprotons by atomic hydrogen and positronium has been simulated by the Classical Trajectory Monte Carlo (CTMC) method. Statistically accurate cross sections for protonium and antihydrogen formation have been obtained and the energy dependence of the process established. Antihydrogen formation from antiproton collisions with positronium in the presence of a laser has also been simulated with the CTMC method and the effects of laser polarisation, frequency and intensity studied. Enhancements of the antihydrogen formation cross section were observed and it is suggested that more sophisticated calculations should be undertaken