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The geometrical concept and structural framework of quasicrystals (QCs) were established shortly after the discovery of QCs. Specifically, the static and dynamic properties of the atoms and the electronic structures of quasiperiodic lattices were theoretically developed. Experimental studies lagged behind theoretical progress due to the lack of suitable samples. This situation changed with the discovery of several highly ordered stable QCs. Studies of these new QCs have provided new insights into the structure and properties of QCs. This article chronologically reviews the discoveries of various stable icosahedral QCs and subsequent studies that determined their structures, properties, and stabilization mechanisms.