  In this paper, we define and study the universal enveloping algebra of
Poisson superalgebras. 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 determine the universal enveloping algebra for examples such as
quadratic polynomial Poisson superalgebras and Poisson symmetric superalgebras.
