  We explicitly calculate the moments $t_n$ of general Heisenberg Hamiltonians
up to eighth order. They have the form of finite sums of products of two
factors. The first factor is represented by a (multi-)graph which has to be
evaluated for each particular system under consideration. The second factors
are well-known universal polynomials in the variable $s(s+1)$, where $s$
denotes the individual spin quantum number. From these moments we determine the
corresponding coefficients of the high-temperature expansion of the free energy
and the zero field susceptibility by a new method. These coefficients can be
written in a form which makes explicit their extensive character. Our results
represent a general tool to calculate eighth-order high-temperature series for
arbitrary Heisenberg models. The results are applied to concrete systems,
namely to magnetic molecules with the geometry of the icosidodecahedron, to
frustrated square lattices, and to the pyrochlore magnets. By comparison with
other methods that have been recently applied to these systems, we find that
the typical susceptibility maximum of the spin-$s$ Heisenberg antiferromagnet
is well described by the eighth-order high-temperature series.
