In this short review, the recently found experimental evidence that Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) states are realized in quasi-two-dimensional (2D) organic superconductors is reported. At low temperatures and when a high magnetic field is aligned parallel to the conducting organic layers, an upturn of the upper critical field much beyond the Pauli limit is observed, as proven by thermodynamic measurements. Under certain conditions, a second thermodynamic transition emerges inside the FFLO state. Nuclear magnetic resonance (NMR) work has added strong microscopic support for the realization of the FFLO state. The NMR spectra in the FFLO phase can very well be explained by a nonuniform one-dimensionally modulated superconducting order parameter. All these features, appearing only in a very narrow angular region close to parallel-field orientation, give robust evidence for the realization of the FFLO state in organic superconductors.