Quasi-two-dimensional organic superconductors are reviewed. These systems exhibit many interesting phenomena, including reduced dimensionality, strong electron - electron and electron - phonon interactions and the proximity of antiferromagnetism, insulator states and superconductivity. Moreover, it has been possible to measure the electronic bands of many of the organics in great detail, in contrast to the situation in other well-known systems in which similar phenomena occur. The crystal structure and normal-state properties of the organics are described before the experimental evidence is presented for and against exotic superconductivity mediated by antiferromagnetic fluctuations. Finally, three instances of field-induced unconventional superconductivity are described.