  A result of Farahat and Higman shows that there is a ``universal'' algebra,
$\mathrm{FH}$, interpolating the centres of symmetric group algebras,
$Z(\mathbb{Z}S_n)$. We explain that this algebra is isomorphic to $\mathcal{R}
\otimes \Lambda$, where $\mathcal{R}$ is the ring of integer-valued polynomials
and $\Lambda$ is the ring of symmetric functions. Moreover, the isomorphism is
via ``evaluation at Jucys-Murphy elements'', which leads to character formulae
for symmetric groups. Then, we generalise this result to wreath products
$\Gamma \wr S_n$ of a fixed finite group $\Gamma$. This involves constructing
wreath-product versions $\mathcal{R}_\Gamma$ and $\Lambda(\Gamma_*)$ of
$\mathcal{R}$ and $\Lambda$, respectively, which are interesting in their own
right (for example, both are Hopf algebras). We show that the universal algebra
for wreath products, $\mathrm{FH}_\Gamma$, is isomorphic to $\mathcal{R}_\Gamma
\otimes \Lambda(\Gamma_*)$ and use this to compute the $p$-blocks of wreath
products.
