Quasiperiodic crystals with long range rotational symmetry but no translational repeat unit have been known in metallic alloys since they were first reported in 1984. Yet only in the past ten years have such complex structures been reported in soft materials, comprised of, e.g., polymers, macromolecules, nanoparticles and colloids. In nearly all of these soft matter systems, quasiperiodicity is entropically stabilized, and any interactions are essentially short range. Interestingly, despite the fact that most metallic quasicrystals exhibit icosahedral symmetry, no icosahedral quasicrystals have been reported for soft matter systems. Instead, primarily 12-fold rotational symmetries are found, with recent, occasional reports of 8-fold, 10-fold, 18-fold, and even 24-fold planar quasicrystals. In this talk, we discuss common features and unifying principles for the self-assembly of soft matter quasicrystals, and we present results for the first icosahedral quasicrystal to be thermodynamically self-assembled in a computer simulation. This icosahedral quasicrystal is robust over a range of parameters, and is obtained from a single particle type interacting via a short-ranged, oscillatory pair potential that may be achievable in systems of colloidal spheres. The icosahedral quasicrystal we report is surrounded in parameter space by clathrates, important for deep sea methane storage, and other new crystal structures never before reported in a one-component system.