  We introduce the analogue of Manin's universal coacting (bialgebra) Hopf
algebra for Poisson algebras. First, for two given Poisson algebras $P$ and
$U$, where $U$ is finite dimensional, we construct a Poisson algebra
$\mathcal{B}(P,\, U)$ together with a Poisson algebra homomorphism
$\psi_{\mathcal{B}(P,\,U)} \colon P \to U \otimes \mathcal{B}(P,\, U)$
satisfying a suitable universal property. $\mathcal{B}(P,\, U)$ is shown to
admit a Poisson bialgebra structure for any pair of Poisson algebra
homomorphisms subject to certain compatibility conditions. If $P=U$ is a finite
dimensional Poisson algebra then $\mathcal{B}(P) = \mathcal{B}(P,\, P)$ admits
a unique Poisson bialgebra structure such that $\psi_{\mathcal{B}(P)}$ becomes
a Poisson comodule algebra and, moreover, the pair $\bigl(\mathcal{B}(P),\,
\psi_{\mathcal{B}(P)}\bigl)$ is the universal coacting bialgebra of $P$. The
universal coacting Poisson Hopf algebra $\mathcal{H}(P)$ on $P$ is constructed
as the initial object in the category of Poisson comodule algebra structures on
$P$ by using the free Poisson Hopf algebra on a Poisson bialgebra (\cite{A1}).
