The mobility of electrons forming the two-dimensional gas in GaN/AlGaN heterostructures is reviewed in connection with recent technological achievements in plasma-assisted molecular beam epitaxy (MBE) growth. We discuss the dependence of the room temperature and liquid helium temperature mobilities on the dislocation density, impurity concentration and compensation. This allows proposing directions of technological developments leading to an improvement of the electron mobility in GaN/AlGaN heterostructures. We also present results on the record mobility observed in GaN/AlGaN heterostructures grown by plasma-assisted MBE on bulk GaN crystals. (c) 2005 Elsevier B.V. All rights reserved.