  $E$-theory was originally defined concretely by Connes and Higson and further
work followed this construction. We generalise the definition to
$C^\ast$-categories. $C^\ast$-categories were formulated to give a theory of
operator algebras in a categorical picture and play important role in the study
of mathematical physics. In this context, they are analogous to
$C^\ast$-algebras and so have invariants defined coming from $C^\ast$-algebra
theory but they do not yet have a definition of $E$-theory. Here we define
$E$-theory for both complex and real graded $C^\ast$-categories and prove it
has similar properties to $E$-theory for $C^\ast$-algebras.
