The Fractional Fourier transform (FRFT) is a relatively novel linear transforms that is a generalization of conventional Fourier transform (FT). FRFT can transform a particular signal to a unified time-frequency domain. In this survey, we try to present a comprehensive investigation of FRFT. Firstly, we provided definition of FRFT and its three discrete versions (weighted-type, sampling-type, and eigendecomposition-type). Secondly, we offered a comprehensive theoretical research and technological studies that consisted of hardware implementation, software implementation, and optimal order selection. Thirdly, we presented a survey on applications of FRFT to following fields: communication, encryption, optimal engineering, radiology, remote sensing, fractional calculus, fractional wavelet transform, pseudo-differential operator, pattern recognition, and image processing. It is hoped that this survey would be beneficial for the researchers studying on FRFT.