  [abridged] We introduce the Phoenix Project, a set of $\Lambda$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 $\sim 130$ 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 three 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 $11 \pm 3%$ of the virial mass
in Phoenix clusters. .
