With demand for aircraft travel set to double in the next twenty years, targets are in place to reduce noise levels and emissions. For example, one target is that the effective perceived noise from aircraft in 2020 should be half of the 2000 level. One of the key noise components is the aeroengine. Building and designing an aeroengine costs millions of pounds and furthermore, to prove the aeroengine is safe, it has to be tested to destruction. Engineers and mathematicians are employed to design aeroengines that will not only be quieter but more fuel efficient and produce fewer harmful emissions while maintaining or improving performance. The main topic of this thesis is investigating rotor-stator interaction which occurs when the turbulent, swirling air produced by the rotor hits the stator and generates noise. We do this in two distinct ways, firstly we calculate the Green’s function for pressure in a turbofan duct with swirling mean flow and secondly we investigate the effect of turbulence hitting an isolated aerofoil. The Green’s function allows engineers to calculate the noise from rotor-stator interaction in simple cases and can be used in beamforming to analyse noise sources in the aeroengine. We consider an infinite duct, and use the Euler equations to derive a sixth order partial differential equation for pressure in the duct. We then find a Green’s function of this equation, which can be done numerically or analytically using high-frequency asymptotics. Our main interest is the analytic Green’s function, which we compare to numerical results. We begin by assuming the base flow has shear and swirling components in a constantly lined duct, and our analytic Green’s function is a new result. We then calculate the Green’s function for a base flow with variable entropy and a lining that varies with circumferential position. To consider flow-blade interaction we simulate the turbulent wake of the rotor hitting a single stator blade. Tests in wind tunnels have shown that, depending on the parameters, introducing a serration on the leading edge of the aerofoil can reduce the noise significantly. We build an analytical model to investigate the effect of the serrated edge, which again involves solving a differential equation by using a Green’s function. It also requires modelling the turbulence, which we do by using either deterministic eddies or stochastic eddies. We show it is possible to reduce the noise by using a serrated leading edge, but it is hard to predict the correct choice of serration to minimise the noise.