While several experimental and numerical studies for Newtonian sprays have been conducted, the exploration of their non-Newtonian counterparts has received comparatively little attention. Achieving a fundamental understanding of the physical phenomena governing spray formation of this type of flow remains a challenge. The numerical simulations of the spray formation of a non-Newtonian fluid still offer substantial challenges, but it is reflective of industrial applications (i.e. spray-drying) and can lead to the optimisation of spray processes containing complex fluids. This thesis aims to provide the basis for the numerical examination of non-Newtonian atomisation and spray systems. We begin with axisymmetric simulations of an impulsively-started viscoelastic jet exiting a nozzle and entering a stagnant gas phase using the open-source code Basilisk. This code allows for efficient computations through an adaptively-refined volume-of-fluid technique that can accurately capture the interface. We use the FENE-P constitutive equation to describe the viscoelasticity of the fluid and employ the log-conformation transformation, which provides stable solutions for the conformation tensor. For the first time, the entire jetting and breakup process of a viscoelastic fluid is simulated, including the flow through the nozzle, which results in an inhomogeneous initial radial stress distribution that affects the subsequent breakup dynamics. The evolution of the velocity field and the elastic stresses in the nozzle are validated against analytical solutions, and the early-stage dynamics of the jet are compared favourably to the predictions of linear stability theory. We explore the effect of flow inside the nozzle on the thinning dynamics of the viscoelastic jet, which develops distinctive "beads-on-a-stringstructures", via analysis of the spatiotemporal evolution of the polymeric stresses. We also systematically investigate the dependence of the filament thinning and breakup characteristics on the axial momentum of the jet and the extensibility of the dissolved polymer chains. We also probe how the secondary droplet formation can be controlled by the finite extensibility of the polymeric chains, as well as the wavenumber of the forced oscillation of the injected liquid at the nozzle inlet. In addition, we study numerically the thinning and breakup in a Dripping-onto-Substrate (DoS) rheometry. The DoS is a conceptually-simple, but dynamically-complex, probe of the extensional rheology of low-viscosity non-Newtonian fluids. It exploits the capillary-driven thinning of a liquid bridge, produced by a single drop as it is dispensed from a syringe pump and spreads laterally onto a solid substrate. By following the filament thinning process, the extensional viscosity and relaxation time of the sample can be determined. Importantly, DoS rheometry allows experimentalists to measure the extensional properties of solutions with lower viscosity than is possible with commercially-available capillary break-up extensional rheometers. Understanding the fluid mechanics underlying the operation of DoS is essential for optimising and extending the performance of this protocol. To achieve this, we employ a computational rheology approach using adaptively-refined axisymmetric numerical simulations with the Basilisk code. The volume-of-fluid technique is used to resolve the moving interface, and the log-conformation transformation provides a stable and accurate solution of the viscoelastic constitutive equation that describes the rheology of the thinning liquid filament. Here, we focus on understanding the role of elasticity and finite chain extensibility in controlling the elasto-capillary (EC) regime, as well as the perturbative effects that gravity and the substrate wettability play in establishing the evolution of the self-similar thinning and pinch-off dynamics. To illustrate the interplay of these different forces, we construct a simple one-dimensional model that captures the initial rate of thinning when the interplay of inertia and capillarity dominates; the model also captures the structure of the transition region to the nonlinear EC regime where the rapidly growing elastic tensile stresses in the thread balance the capillary pressure as the filament thins towards breakup. Finally, we develop and test a rheological model for avoiding the numerical challenges associated with the commonly-used constitutive equations for viscoelastic extensional flows, which accounts for the changes in the fluid viscosity based on the principal invariants of the deviatoric stress tensor. We validate the predictions of the model against a free-filament thinning and a jetting flow configuration of a FENE-P fluid, highlighting its capability to account for a substantial increase in viscosity under elongation. The model, however, fails to exhibit all of the characteristic viscoelastic flow regimes observed in our FENE-P-based simulation results. This highlights the need for further model improvement incorporating the flow kinematics history, a distinctive characteristic of viscoelasticity, which will be the subject of future work.