  We introduce one of the most beautiful algebraic varieties known, a quintic
hypersurface in projective five-space, which is invariant under the action of
the Weyl group of $E_6$. This variety is intricately related with many other
moduli problems, some of which are: marked hyperelliptic curves of genus two,
Picard curves of genus four with a $\sqrt{-3}$-level structure, six points on
the projective line, abelian surfaces with (1,3) polarisations, quartic
surfaces invariant under the action of the Heisenberg group in projective
three-space, K3-surfaces which are double covers of the projective plane
branched along six lines, and last but not least, cubic surfaces in projective
three-space. These relationships are developed in some detail, with particular
care on the birational aspects.
