Forty years after the discovery of the first organic superconductor, the nature of the superconducting state in these materials is still not fully understood. Here, I present an overview on the historical developments and current knowledge on this topic for the quasi-one- and quasi-two-dimensional (2D) organic charge-transfer salts. Thereby, I focus on the prototype materials based on the donor molecules tetramethyltetraselenafulvalene (TMTSF) and bisethylenedithio-tetrathiafulvalene (BEDT-TTF or ET for short). 2D organic superconductors based on the latter molecule are found to show Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) states at high magnetic fields and low temperatures. Thermodynamic and nuclear magnetic resonance data give robust evidence for the existence of this FFLO state with modulated order parameter.