This thesis makes use of the intermediate energy R-matrix approach (IERM) in computationally modelling various single and double ionisation processes. The basis used in the IERM approach is an ideal candidate for representing a double electron continua in a time-dependent or time-independent, general N-electron R-matrix code and one of the main objectives of the thesis is to explore the efficacy of the basis used in the IERM approach in representing a double-electron continua. To do this, the IERM method is applied to the study of photoionisation and photo-double-ionisation of various He-like and Be-like atoms and ions in the ground and excited initial states, and preliminary calculations of electron impact excitation involving high lying energy levels of H is investigated with the two-dimensional R-matrix propagator package. The convergence of the basis for these processes is examined. Throughout the thesis the results obtained with the IERM method are compared to experimental results and those from other state of the art theoretical approaches, and agreement is excellent in all cases. In the case of He-like and Be-like atoms and ions, trends in moving along the isoelectronic sequences and between the ground and various excited states are analysed and compare well with existing models where available. The data presented is also of use in astrophysical applications, to experimentalist and to other theorist who wish to study these or similar systems. The IERM basis is shown to be capable of representing a double electron continua and directions in which further work in this field could take are highlighted.