  In this thesis, we consider a notion of a higher version of the relation
between Courant-Dorfman algebras and Poisson vertex algebras. We define a
higher Courant-Dorfman algebra, and study the relationship with graded
symplectic geometry. In particular, we give graded Poisson algebras of degree
$-n$ in the non-degenerate case. For higher Courant-Dorfman algebras coming
from finite-dimensional vector bundles, they coincide with the algebras of
functions of the associated differential-graded(dg) symplectic manifolds of
degree $n$.
  We define a higher Lie conformal algebra and Poisson vertex algebra, and give
a higher (weak) Courant-Dorfman algebraic structure arising from them.
Moreover, we prove that the higher Lie conformal algebras and higher Poisson
vertex algebras have properties like Lie conformal algebras and Poisson vertex
algebras. As an example, we obtain an algebraic description of
Batalin-Fradkin-Vilkovisky(BFV) current algebras.
