Examples of stable binary icosahedral quasicrystals are relatively rare, and at present there are no known examples featuring localized magnetic moments. These would represent an ideal model system for attaining a deeper understanding of the nature of magnetic interactions in aperiodic lattices. Here we report the discovery of a family of at least seven rare earth icosahedral binary quasicrystals, i-R-Cd (R = Gd to Tm, Y), six of which bear localized magnetic moments. Our work highlights the importance of carefully motivated searches through phase space(1) and supports the proposal that, like icosahedral Sc12Zn88 (ref. 2), binary quasicrystalline phases may well exist nearby known crystalline approximants, perhaps as peritectically forming compounds with very limited liquidus surfaces, offering very limited ranges of composition/temperature for primary solidification.