The Stockwell transform (ST), which is an invertible time-frequency spectral localization technique, uniquely combines elements of wavelet transform and short-time Fourier transform. However, its signal analysis ability is restricted in the time-frequency plane. In this paper, the novel fractional Stockwell transform (FRST) is proposed to address this problem, based on the ST and fractional Fourier transform (FRFT). It displays the time and fractional-frequency information jointly in the time-fractional-frequency plane. More importantly, it has explicit physical interpretation and usefulness for practical applications. First, the theories of continuous FRST are presented in detail, including its definition, basic properties and time-fractional-frequency analysis. Next, the discrete version for the transform, together with its reconstruction formula is considered. Further, the FRST in the realm of almost periodic functions is explored to expand its theory in the persistent signal space. Lastly, we investigate and discuss several applications based on FRST, including the time-frequency of chirp signals and the detection of disturbed chirp signals. Simulations are presented to verify the rationality and effectiveness of the FRST.