  In this paper, we define and study the universal enveloping algebra of a
Poisson superalgebra. In particular, a new PBW Theorem for Lie-Rinehart
superalgebras is proved leading to a PBW Theorem for Poisson superalgebras, we
show the universal enveloping algebra of a Poisson Hopf superalgebra (resp.
Poisson-Ore extension) is a Hopf superalgebra (resp. iterated Ore extension),
and we study the universal enveloping algebra for interesting classes of
Poisson superalgebras such as Poisson symplectic superalgebras.
