Intense, ultra-short, laser-atom interactions have seen a lot of interest in recent decades due to the continuous development of ever more powerful sources, providing Extreme Ultra-Violet (XUV) and X-ray pulses whose peak potentials are close to if not on par with the core potentials of the lighter atoms such as Li or Mg. In such scenarios, theories that assume the laser field to be negligible in relation to the atomic core potential are no longer applicable, in turn, this regime requires an ab-initio treatment, one that involves directly solving the Time-Dependent Schrödinger Equation (TDSE) of the system. Such a formulation can quickly become complex, depending on the chosen approximations such as the choice of basis, and the implementation of the laser field. The work that constitutes this thesis provides two key innovations: (a) a novel theoretical treatment of the two-electron TDSE, which involves direct time propagation on the uncorrelated basis of direct products of one-electron states; and (b) a modern computation framework for the solution of both the one- and two-electron TDSE. The former functions as an alternative to the conventional Configuration Interaction (CI) picture, this uncorrelated basis – while being more computationally demanding when it comes to time propagation – is more representative of the doubly ionised system. The latter contribution consists of a modern, open source, suite of C++ programs for the solution of the one- and two-electron TDSE for workstations and desktop computers. These innovations expand the theoretical and computational coverage of the two-electron TDSE problem. Techniques learned during the code development were used to simulate experimental schemes in neon targets under attosecond radiation, while later the developed code was utilised to obtain results in atomic hydrogen and helium.