  Jet manifolds and vector bundles allow one to employ tools of differential
geometry to study differential equations, for example those arising as
equations of motions in physics. They are necessary for a geometrical
formulation of Lagrangian mechanics and the calculus of variations. It is thus
only natural to require their generalization in geometry of $\mathbb{Z}$-graded
manifolds and vector bundles.
  Our aim is to construct the $k$-th order jet bundle
$\mathfrak{J}^{k}_{\mathcal{E}}$ of an arbitrary $\mathbb{Z}$-graded vector
bundle $\mathcal{E}$ over an arbitrary $\mathbb{Z}$-graded manifold
$\mathcal{M}$. We do so by directly constructing its sheaf of sections, which
allows one to quickly prove all its usual properties. It turns out that it is
convenient to start with the construction of the graded vector bundle of $k$-th
order (linear) differential operators $\mathfrak{D}^{k}_{\mathcal{E}}$ on
$\mathcal{E}$. In the process, we discuss (principal) symbol maps and a
subclass of differential operators whose symbols correspond to completely
symmetric $k$-vector fields, thus finding a graded version of Atiyah Lie
algebroid. Necessary rudiments of geometry of $\mathbb{Z}$-graded vector
bundles over $\mathbb{Z}$-graded manifolds are recalled.
