The focus of this thesis is to understand a variety of emergent materials properties that are driven by quantum phenomena. A range of materials are investigated where the crystal, electronic and magnetic structures are delicately coupled to give rise to intriguing macroscopic properties. Computational methods are the primary tool for tackling these problems. This is largely centred around quantum mechanical, ab initio calculations using density functional theory (DFT), but also includes mean-field analyses and stochastic simulations of a model Hamiltonian. Tin telluride, SnTe, is a crystalline topological insulator with potential applications in a new generation of spintronics devices. In practice, SnTe shows a low temperature ferroelectric distortion and contains a large number of bulk carriers from Sn vacancies, so stands as a rare example of coexistence between metallicity and ferroelectricity. The implications of these effects for topological and transport properties require accurate modelling of the electronic structure. Here, in close collaboration with experiment, the evolution of the Fermi surface across this structural transition is probed by calculating quantum oscillation frequencies. The image analysis tools of Mathematica were exploited to develop a code for computing these frequencies from DFT electronic structure calculations. Agreement between experiment and theory is crucially dependent on the crystal structure. Calcium ruthenate, Ca2RuO4, is a layered perovskite compound with a rich phase diagram, which can be traced back to its strongly correlated electronic structure. Hydrostatic pressure can be used as a means to manipulate its crystal structure, which encourages several interesting effects. In particular, Ca2RuO4 undergoes anomalous expansion of the c axis, and a first-order structural transition coupled with a Mott insulator-metal transition. Here, this pressure response is investigated with DFT+U calculations, which account for the importance of electron correlation by adding an on-site Hubbard-like repulsion term. This work presents the first fully self-consistent electronic structure for Ca2RuO4, obtained from optimised crystal structures along a sequence of pressures. This appreciation of the coupling between lattice and electronic degrees of freedom sheds some light on its unusual phase diagram. The insulator-metal transition is reproduced and naturally coincides with a structural transition and associated orbital order. For the metallic phase, a complex energetic landscape with several competing phases emerges. Uranium gold, UAu2, is a heavy fermion metal with a complex spin-density-wave (SDW) phase at low temperature. This phase consists of frustrated, incommensurate ordering that is very robust in external fields, but suppressed by pressure to reveal unconventional superconductivity. The origin of this exotic magnetic ordering is of interest here, which is explored by several different approaches with DFT+U. Both itinerant and local-moment pictures of the magnetism are entertained. Fermi surface nesting is shown to be ineffective, but mapping the system to a Heisenberg model and computing effective exchange interactions identifies an instability towards modulated order. In addition to these material-specific investigations, the treatment of longitudinal magnetic fluctuations in computational methods is studied. Magnetic fluctuations are important for understanding materials on the border between itinerant and local-moment magnetism. The continuous-spin Ising (CSI) model is investigated here as a phenomenological model of these fluctuations. Using a bespoke simulation technique this model is extensively explored, firstly on a cubic ferromagnet and secondly on a highly frustrated, stacked triangular lattice with a variety of interaction topologies. The introduction of fluctuations is shown to alter the ground state of the prototypical frustrated triangular lattice, and to enhance the transition temperature of more severely frustrated systems. In all of these cases, different degrees of freedom in the system couple to one another, either to make an accurate theoretical description difficult, or to give rise to unexpected emergent properties. Both SnTe and Ca2RuO4 represent examples of delicately coupled crystal and electronic structures. In SnTe, the tuning of the crystal structure is vital to correctly describe the Fermi surface, while in Ca2RuO4 the Ru orbital structure very directly determines the structural properties, and drives the unusual pressure response. UAu2 introduces a non-trivial magnetic structure, which requires careful treatment of electron-electron interactions to locate. In the study of the CSI model, altering the interaction topology, which acts as a proxy for manipulating the underlying electronic structure, has drastic implications for the role of magnetic fluctuations in the system.