This paper reviews a feature of atomically-clean quasicrystal surfaces that distinguishes them from surfaces of crystalline materials. That feature is a high degree of heterogeneity among different terraces, and among structurally-identical adsorption sites. The heterogeneity can be both structural and chemical in origin. A large variability is expected even for a surface which is perfectly bulk-terminated, and we call this intrinsic heterogeneity. Additional variability can derive from the surface preparation process, which can yield metastable structures. We call this extrinsic heterogeneity. Experimental evidence is given for both cases. This heterogeneity can be an important factor in understanding and predicting surface phenomena such as chemisorption.