This thesis describes the development of two numerical models for the study of (1) incompressible multiphase flow and (2) compressible multiphase flow. Both models employ a state-of-the-art adaptive unstructured mesh-based approach which allows the mesh, upon which the model equations are discretised, to be optimised in order to focus numerical resolution in areas important to the dynamics and decrease it where it is not needed as a simulation progresses. The implementation of the models takes place within a computational fluid dynamics code called Fluidity. The application of the models concerns the multi-scale simulation of volcanic ash transport in aqueous solutions and in the atmosphere. Simulations of ash settling in a water tank, which mimic published laboratory experiments, are performed primarily in two dimensions. The results demonstrate that ash particles can either settle slowly and individually, or rapidly and collectively as an ash-laden cloud, referred to as a plume. Two quantities used to measure the tendency for plumes to form are then evaluated with a parameter study. Particles settling collectively are slowed by inertial drag, rather than viscous drag, and it is shown that such quantities must account for this. An improvement to the measures is proposed, along with an alternative measure which uses a more accurate expression for the collective settling timescale. Finally, a two-dimensional kilometre-scale volcanic eruption of hot gas and ash into the atmosphere is simulated. The results are compared with those from MFIX, a leading multiphase flow code. Both Fluidity and MFIX are able to successfully capture the key characteristics of an eruption event. The benefits of the adaptive unstructured mesh-based approach are highlighted throughout the thesis by demonstrating that it maintains solution accuracy whilst providing a substantial reduction in computational requirements when compared to the same simulation performed using a fixed mesh of uniform resolution.