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A new approach to accelerating the evaluation of monopole and dipole source integrals via the fast multipole method (FMM) in the time domain for general three-dimensional (3-D) aeroacoustic problems is presented in this paper. In this approach, the aeroacoustic field is predicted via a hybrid method that uses computational fluid dynamics (CFD) for near-field flow field calculations and the Ffowcs Williams–Hawkings (FW-H) acoustic analogy for far-field sound field predictions. The evaluation of the surface integrals of the monopole and dipole source terms appearing in the FW-H formulation is accelerated by a 3-D FMM to reduce computational cost. The proposed method is referred to as Fast FW-H in this work. The performance and efficiency of the proposed methodology are demonstrated using several examples. First, aeroacoustic predictions for the cases of a stationary acoustic monopole, moving acoustic monopole and stationary acoustic dipole in a uniform flow are studied, generally showing good agreement with the analytical solutions. Second, the sound field radiating from a flow passing a finite-length circular cylinder and the propeller of an unmanned aerial vehicle (UAV) during forward flight are studied, and the computed results obtained via the FW-H and Fast FW-H methods in the time domain with a stationary, permeable surface are compared. The overall computational efficiency of the sound field solutions obtained via the Fast FW-H method is found to be approximately two times faster than the computational efficiency of the original FW-H method, indicating that this proposed approach can be an accurate and efficient computational tool for modelling far-field aeroacoustic problems.