  In this paper, we introduce the notions of differential graded (DG) Poisson
algebra and DG Poisson module. Let $A$ be any DG Poisson algebra. We construct
the universal enveloping algebra of $A$ explicitly, which is denoted by
$A^{ue}$. We show that $A^{ue}$ has a natural DG algebra structure and it
satisfies certain universal property. As a consequence of the universal
property, it is proved that the category of DG Poisson modules over $A$ is
isomorphic to the category of DG modules over $A^{ue}$. Furthermore, we prove
that the notion of universal enveloping algebra $A^{ue}$ is well-behaved under
opposite algebra and tensor product of DG Poisson algebras. Practical examples
of DG Poisson algebras are given throughout the paper including those arising
from differential geometry and homological algebra.
