The fractional Fourier transform (FRFT) is a generalization of the Fourier transform. The FRFT can characterize signals in multiple fractional domains and provide new perspectives for non-stationary signal processing and linear time variant system analysis, thus it is widely used in reality applications. We first review recent developments of the FRFT in theory, including discretization algorithms of the FRFT, various discrete fractional transforms, sampling theorems in fractional domains, filtering and parameter estimation in fractional domains, joint analysis in multiple fractional domains. Then we summarize various applications of the FRFT, including radar and communication signal processing in fractional domains, image encryption, optical interference measurement, medicine, biology, and instrument signal processing based on the FRFT. Finally we discuss the future research directions of the FRFT, including fast algorithm of the FRFT, sparse sampling in fractional domains, machine learning utilizing the FRFT, graph signal processing in fractional domains, and discrete FRFT based on quantum computation.