  We present a new $hp$-version space-time discontinuous Galerkin (dG) finite
element method for the numerical approximation of parabolic evolution equations
on general spatial meshes consisting of polygonal/polyhedral (polytopic)
elements, giving rise to prismatic space-time elements. A key feature of the
proposed method is the use of space-time elemental polynomial bases of
\emph{total} degree, say $p$, defined in the physical coordinate system, as
opposed to standard dG-time-stepping methods whereby spatial elemental bases
are tensorized with temporal basis functions. This approach leads to a fully
discrete $hp$-dG scheme using less degrees of freedom for each time step,
compared to standard dG time-stepping schemes employing tensorized space-time,
with acceptable deterioration of the approximation properties. A second key
feature of the new space-time dG method is the incorporation of very general
spatial meshes consisting of possibly polygonal/polyhedral elements with
\emph{arbitrary} number of faces. A priori error bounds are shown for the
proposed method in various norms. An extensive comparison among the new
space-time dG method, the (standard) tensorized space-time dG methods, the
classical dG-time-stepping, and conforming finite element method in space, is
presented in a series of numerical experiments.
