FFT methods have become a fundamental tool in computational micromechanics since they were first proposed in 1994 by Moulinec and Suquet for the homogenization of composites. Since then many different approaches have been proposed for a more accurate and efficient resolution of the non-linear homogenization problem. Furthermore, the method has been pushed beyond its original purpose and has been adapted to a variety of problems including conventional and strain gradient plasticity, continuum and discrete dislocation dynamics, multi-scale modeling or homogenization of coupled problems such as fracture or multi-physics problems. In this paper, a comprehensive review of FFT approaches for micromechanical simulations will be made, covering the basic mathematical aspects and a complete description of a selection of approaches which includes the original basic scheme, polarization based methods, Krylov approaches, Fourier–Galerkin and displacement-based methods. Then, one or more examples of the applications of the FFT method in homogenization of composites, polycrystals or porous materials including the simulation of damage and fracture will be presented. The applications will also provide an insight into the versatility of the method through the presentation of existing synergies with experiments or its extension toward dislocation dynamics, multi-physics and multi-scale problems. Finally, the paper will analyze the current limitations of the method and try to analyze the future of the application of FFT approaches in micromechanics. [ABSTRACT FROM AUTHOR]