This thesis is focused on the development of an efficient first-principles theoretical and numerical method based on the many-electron algebraic diagrammatic construction [ADC(n)] schemes, in order to describe the correlated ionisation dynamics in atomic and molecular systems interacting with perturbative and non-perturbative laser fields. The first line of research has focused on the calculation of total single-photon photoionisation cross-sections, applying the Stieltjes-Imaging theory to Lanczos pseudospectra of the ADC Hamiltonian in Gaussian basis. We have established the accuracy of this technique by comparing the ADCLanczos-Stieltjes ground-state cross-sections obtained using different levels of many-body theory to the experimental ones for a series of organic molecules. We have extended this method to excited states cross-sections showing that a theoretical modelling of photoionisation from excited states requires an intrinsically double excitation theory. However, above 80 eV photon energy all three methods lead to inaccurate results due to the limitations of the Gaussian basis to describe continuum wave-functions of ionised electrons. The second, main line of research, has therefore been dedicated to constructing and computationally optimising the first implementation of the single [ADC(1)] and double excitations [ADC(2)] schemes in the B-spline basis, which is able to accurately describe the strongly oscillating continuum orbitals. As first application of this new method, we have calculated the photoionisation cross-sections of noble gas atoms showing that the features that pose a challenge for the GTO calculations are reproduced in a very good agreement with the experiment. We also have developed a time-dependent version with which we have calculated the HHG spectra of Ar, reproducing the effect of the Cooper minimum, and CO2, quantitatively investigating the multi-channel effects on its dynamical minimum. Finally we have provided a numerical answer to the highly topical question of coherence and ionic wavepacket formation in short pulse photoionisation.