The structure of the hardening η′-phase precipitates in Al–Zn–Mg alloys has been investigated by using precession electron diffraction and X-ray synchrotron radiation diffraction. The latter was recorded as a three-dimensional, continuous intensity distribution from a single alloy grain, from which patterns of remarkable sharpness from the precipitate particles could be extracted. High resolution electron microscopy revealed extensive structural variations with prolific faults in the η′ precipitates. The η′-structures are described in terms of two structure models based on two icosahedral elements that are inherent in the equilibrium η-MgZn2 structure. The role of these icosahedra in the transformations of the alloy system is discussed.