Viscoplastic fluids are a class of non-Newtonian liquids characterised by their yield stress. Unless an external stress is applied which is larger than this threshold value, the fluid does not flow, but exhibits rigid body behaviour. Above the yield stress, applied forces cause viscous deformation. Such fluids play important roles in a range of fields, notably in wellbore drilling, which is the application that motivated this project. One aspect of this operation requires displacement of drilling fluid by cement in the annulus between casing and geological surroundings, and both of these fluids are viscoplastics. Ensuring that this is done properly is of utmost importance to the overall safety of the drilling operation. Often, numerical simulations are the only viable way of experimenting with the effect of drilling parameters and fluid properties on the flow configuration and resulting behaviour. Unfortunately, the presence of a yield stress leads to a singularity in the apparent viscosity at zero strain. This causes substantial computational expense for the algorithms used to simulate fluid flow numerically, even when regularisation techniques are employed to alleviate the problem. Consequently, most published results in the literature on computational viscoplasticity has been restricted to two-dimensional and steady-state flows. In an attempt to address this, we have applied state-of-the-art techniques from high-performance computational fluid dynamics to the viscoplastic flow problem. Specifically, we utilise spatio-temporal adaptive mesh refinement on structured meshes in this context for the first time. This is achieved through the software framework AMReX, which includes state-of-the-art numerical tools for solving partial differential equations with optimal parallel scaling. The ability to rapidly simulate unsteady viscoplastic flow problems in three dimensions is demonstrated by novel numerical experiments in a lid-driven cavity. In order to investigate flows in more interesting domain geometries and around objects, an embedded boundary algorithm has been developed which works alongside the viscoplastic flow solver. We show how this methodology can be utilised to simulate flow inside non-rectangular objects, and investigate fully three-dimensional viscoplastic flow past several shapes of bodies for the first time.