Synthetic jet actuators have previously been defined as having potential use in both internal and external aerodynamic applications. The formation of a jet flow perpendicular to the surface of an aerofoil or in a duct of diffuser has a range of potential flow control benefits. These benefits can include both laminar to turbulent transition control, which is associated with a drag reduction in aerodynamic applications. The formation and development of zero-net-mass-flux synthetic jets are investigated using highly accurate numerical methods associated with the methodology of Direct Numerical Simulation (DNS). Jet formation is characterised by an oscillating streamwise jet centreline velocity, showing net momentum flux away from the jet orifice. This momentum flux away from the orifice takes the form of a series of vortex structures, often referred to as a vortex train. Numerical simulations of the synthetic jet actuator consist of a modified oscillating velocity profile applied to a wall boundary. The Reynolds numbers used vary from 85 ≤ Re ≤ 300. A complete numerical study of both axisymmetric and fully three-dimensional jet flow is performed. A parametric axisymmetric simulation is carried out in order to study the formation criterion and evolution of zero-net-mass-flux synthetic jets under variations in actuator input parameters. From the results of these simulations the conditions necessary for the formation of the synthetic jet along with the input parameters that provide an optimal jet output are deduced. Jet optimisation is defined by the mass flow, vortex strength and longevity of the vortex train as it travels downstream. Further investigations are carried out on a fully three-dimensional DNS version of the optimised axisymmetric case. Comparisons between the jet evolution and flow-field structures present in both the axisymmetric and three-dimensional configurations are made. This thesis examines the vortex structures, the jet centreline velocities along with time dependent and time averaged results in order to deduce and visualise the effects of the input parameters on the jet formation and performance. The results attained on altering the oscillation frequency of the jet actuator indicated that synthetic jets with zero mean velocity at the inflow behave significantly differently from jets with non-zero mean velocity at the inflow. A study into the evolution and formation of the train of vortex structures associated with the formation of a synthetic jet is performed. This study is accompanied with a series of time averaged results showing time dependent flow-field trends. The time history of the jet centreline velocity, showing the net momentum flux of the fluid away from the orifice of a fully developed synthetic jet, is analysed for both axisymmetric and three-dimensional cases. Differences in the fluid dynamics between the idealised axisymmetric configuration and the three-dimensional case have been identified, where three-dimensional effects are found to be important in the region near the jet nozzle exit. The effect of a disturbance introduced into the three-dimensional simulation in order to break its inherent symmetr around the jet centreline is examined by altering the input frequency of the disturbance. It was found that the effect of this relatively minor disturbance had a major effect on the jet flow field in the region adjacent to the orifice. The effect of which was deemed to be caused by discontinuities in the surface of the jet orifice due to manufacturing tolerances. Although the effects of these disturbances on the jet flow-field are large, they seem to have been neglected from numerical simulations to date. The effect of a synthetic jet on an imposed cross-streamwise velocity profile was examined. It was found that the synthetic jet flow-field resulted in a deformation of the velocity profile in the region downstream of the synthetic jet. It is suggested that this region of deformed flow could interact with coherent structures in a transitional boundary layer in order to delay flow transition to turbulence. The effect of varying the Strouhal number of a synthetic jet in a cross-flow is also analysed. It is clear from the results presented that, in the presence of a cross-flow velocity the Strouhal number effect on the synthetic jet flow field evolution, while dominant in a quiescent fluid is surpassed by the effect of the cross-flow.