An important criterion in the development of modern aeroengines is the identification of the dominant noise sources under typical aircraft take-off and approach conditions, and also in ground-based tests in which the engine is stationary. In this paper, we develop a theoretical model for unsteady distortion noise, which results from the interaction of ingested atmospheric turbulence with the rotating fan, with a view to providing a better understanding of the important physical mechanisms in this particular aspect of sound generation. The theory, developed in the frequency domain, is applicable for any arbitrary spectral form of atmospheric turbulence upstream of the fan, and as a simple model we take the von Kármán spectra for isotropic turbulence. The key fluid dynamical process in unsteady distortion is the deformation of turbulent eddies into long, narrow filaments as they enter the engine, due to the strong streamtube contraction experienced by the steady, non-uniform mean flow generated by the fan. Simple models of the steady flow fields are provided for both open and ducted rotor geometries. The distorted turbulent field at the fan face can be obtained using rapid distortion theory, and considerable simplification is made here by noting that the number of blades in typical aeroengine fans is large, allowing the application of asymptotic analysis and the derivation of closed-form expressions for those parts of the turbulence spectrum at the fan face which dominate the radiation. The unsteady forces exerted on the rotating fan blades are then calculated via a strip-theory approach. The resulting sound scattered to the far field is then evaluated using asymptotic theory for open and ducted rotors. Results are presented in the form of frequency spectra for the turbulent field at the fan face, the blade forces and the radiated sound for typical testing and aircraft operating conditions. High tonal noise levels are obtained under static conditions, whereas the sound is generally broadband in flight. The dependence on turbulence parameters such as the integral lengthscale is highlighted.