This is a progress review of an emerging research front: soft quasicrystals including liquid crystalline dendrons, nanoparticles, mesoporous silica, colloids, ABC star and linear terpolymers, and even water and silicon. As an aid to readers, we explain the basics of quasicrystals developed in solid-state physics: orders in quasicrystals, higher dimensional crystallography, approximants, phason randomness, and the origin of quasicrystal formation. Then we review some numerical studies from early to recent ones. Our main purpose is to elucidate how to construct quasicrystalline structures: The introduction of additional components or a new length-scale is the key to discover new quasicrystals. As a case study, we describe our recent studies on ABC star terpolymer systems and present the results of simulations of dodecagonal polymeric quasicrystals. In the case of dodecagonal quasicrystals, one easily finds that the key is to search square-triangle tiling structures with changing components. Application to photonic quasicrystals is reviewed as well. Our hope is that this review will contribute furthering quasicrystal chemistry. (C) 2011 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 50: 155-167, 2012