We study excited-state phenomena in a variety of semiconductor systems, with use of the variational and diffusion quantum Monte Carlo (QMC) methods. Firstly, we consider the formation of charge-carrier complexes in the Mott-Wannier model, for systems of restricted geometrical freedom (the coupled quantum well bilayer, and the quantum ring). We find in such systems that geometrical constraints lead to the characteristic formation of certain charge-carrier complexes, and highlight how such effects are of relevance to the interpretation of recent experiments. Secondly, we illuminate a key difference between two-dimensional systems formed from geometrical restriction, and those which are truly two-dimensional in extent, by introduction of the Keldysh interaction. We then study the formation of charge-carrier complexes in two-dimensional semiconductors and their heterostructures in the so-called Mott-Wannier-Keldysh model, deriving appropriate extensions of the Keldysh interaction as necessary. Thirdly, we undertake a comprehensive survey of the use of continuum QMC methods to evaluate excited-state properties in a truly ab initio fashion, establishing best-practices, and presenting energy gap calculations for several real materials. This includes the first published QMC calculation of the electronic energy gaps of a two-dimensional semiconductor, phosphorene. Finally, we propose an extension of the Keldysh interaction which permits the study of continuum phases, the so-called ``periodic Keldysh interaction'', and use it to probe the possible Wigner crystallisation of electrons in a weakly-doped two-dimensional semiconductor
