We introduce the Phoenix Project, a set of Λ cold dark matter (CDM) simulations of the dark matter component of nine rich galaxy clusters. Each cluster is simulated at least at two different numerical resolutions. For eight of them, the highest resolution corresponds to ∼130 million particles within the virial radius, while for one this number is over one billion. We study the structure and substructure of these systems and contrast them with six galaxy-sized dark matter haloes from the Aquarius Project, simulated at comparable resolution. This comparison highlights the approximate mass invariance of CDM halo structure and substructure. We find little difference in the spherically averaged mass, pseudo-phase-space density and velocity anisotropy profiles of Aquarius and Phoenix haloes. When scaled to the virial properties of the host halo, the abundance and radial distribution of subhaloes are also very similar, despite the fact that Aquarius and Phoenix haloes differ by roughly three decades in virial mass. The most notable difference is that cluster haloes have been assembled more recently and are thus significantly less relaxed than galaxy haloes, which leads to decreased regularity, increased halo-to-halo scatter and sizable deviations from the mean trends. This accentuates the effects of the strong asphericity of individual clusters on surface density profiles, which may vary by up to a factor of 3 at a given radius, depending on projection. The high apparent concentration reported for some strong-lensing clusters might very well reflect these effects. A more recent assembly also explains why substructure in some Phoenix haloes is slightly more abundant than in Aquarius, especially in the inner regions. Resolved subhaloes nevertheless contribute only 11 ± 3 per cent of the virial mass in Phoenix clusters. Together, the Phoenix and Aquarius simulation series provide a detailed and comprehensive prediction of the CDM distribution in galaxies and clusters when the effects of baryons can be neglected.