  We introduce the notion of a $\hbar$-vertex algebra, which is an algebraic
structure similar to a vertex algebra, with a deformed translation covariance
axiom. We study the structure theory of $\hbar$-vertex algebras, highlighting
similarities with the structure theory of vertex algebras. We prove results
analogous to Goddard's Uniqueness Theorem, Reconstruction Theorem, Borcherds
Identity and OPE Expansion Formula. We also introduce the related notions of a
$\hbar$-Lie conformal algebra and of a $\hbar$-Poisson vertex algebra. The main
example of $\hbar$-vertex algebras are vertex algebras with deformed vertex
operators $Y_\hbar(a,z)=(1+\hbar z)^{\Delta_a}Y(a,z)$, where $\Delta_a$ is the
eigenvalue of a Hamiltonian operator $H$. We simplify the construction of the
Zhu algebra using the formalism of $\hbar$-vertex algebras. As a further
application of the theory developed in the first part of the paper, we discuss
the chiralization of classical star-products. We compute explicit formulae for
the chiralization of a class of star-products, including the Moyal-Weyl and the
Gutt star-products. Putting $\hbar=0$, these formulae yield explicit formulae
for deformation quantizations of a class of Poisson vertex algebras.
