THE attainment of unprecedentedly high transition temperatures (T(c)s) in the copper oxide superconductors illustrates how working with more complex chemical systems allows greater opportunity to balance opposing forces within a single chemical compound, leading to a better optimization of physical properties. For many desired properties, materials with optimal chemical complexity have undoubtedly not yet been found. This appears to be the case for the intermetallic superconductors, whose study has languished in recent years, and which almost never show T(c)s above 15 K. These are almost all binary compounds with substitution-type additives, or, rarely, true ternary compounds such as LuRh4B4 (T(c) = 11.7 K; refs 1, 2). If, as some argue (refs 3, 4), materials such as A(x)C60 (ref. 5) and Ba0.6K0.4BiO3 (refs 6, 7) are conventional electron-phonon superconductors with T(c)s of approximately 30 K, then the absence of higher T(c)s in intermetallic compounds may mean only that more complex materials have not been sufficiently explored. We have recently found superconductivity at 23 K (a T(c) equal to that of the previous intermetallic record holder, Nb3Ge; ref. 9) in the quaternary intermetallic system yttrium-palladium-boron-carbon8, but we were unable to identify the superconducting phase. Here we report superconductivity at temperatures up to 16.6 K for the single-phase quaternary intermetallic compounds LnNi2B2C (where Ln stands for Y, Tm, Er, Ho or Lu). The presence of the 3d transition metal nickel, and the layered crystal structure10 raise intriguing questions about the origin of the superconductivity, and the relatively high T(c)s of these and the Y-Pd-B-C superconductor suggest that there may yet be more surprises in store.