The chemical vapour deposition (CVD) of graphene is the most promising route for production of large-area graphene films. However there are still major challenges faced by the field, including control of the graphene coverage, quality, and the number of layers. These challenges can be overcome by developing a fundamental understanding of the graphene growth process. This thesis contributes to the growing body of work on graphene CVD by uniquely exploring the gas phase chemistry and fluid flow in the hot-wall graphene CVD reactor. Firstly the reported parameter space for the hot-wall CVD of graphene on copper was mapped, informing the subsequent work and providing a resource for the wider community. A CVD reactor was constructed to extend this parameter space to lower pressures using methane as a carbon source, and the films were categorised using scanning electron microscopy, Raman spectroscopy and optical dark field microscopy. The latter showed particular promise as a rapid and non-destructive characterization technique for identifying graphene films on the deposition substrate. The gas phase equilibrium compositions were calculated across the parameter space, and correlations between the stabilities of various chemical species and the types of deposition were drawn. This laid a foundation for the remainder of the experimental work, which explored the effect of diluent gases and different feedstocks on the growth to understand the importance of the identified correlations. Diluent gases (argon and nitrogen) were added to the experimental conditions and the thermodynamic model, and were found to reduce the degree of coverage of the graphene films. This result shows that the CVD of graphene is sensitive to factors other than the thermodynamic state parameters, such as the fluid flow profile in the reactor and inelastic collisions between the higher mass diluent gases and the methane/hydrogen/copper system. Using a nitrogen diluent raises the equilibrium carbon vapour pressure and seems to allow larger graphene grains to form. This suggests that thermodynamic factors can contribute to the nucleation of graphene films. Varying the hydrocarbon feedstock and the process conditions indicated that the structure of the deposited carbon is closely related to the nucleation kinetics. Three nucleation regimes are associated with different types of deposition: homogeneous nucleation with amorphous carbon or soot; uncatalysed nucleation with multilayer deposition; and nucleation processes controlled by the copper substrate withpredominantly monolayer deposition. Changing the feedstock from methane to acetylene resulted in poorer graphene coverage, showing that thermodynamic control does not apply in the portion of the parameter space at the high temperatures and lowpressures most successfully used for the deposition of continuous graphene monolayers.