  For a connected reductive group $G$ over ${\mathbb R}$, we study
cohomological $A$-parameters, which are Arthur parameters with the
infinitesimal character of a finite-dimensional representation of $G({\mathbb
C})$. We prove a structure theorem for such $A$-parameters, and deduce from it
that a morphism of $L$-groups which takes a regular unipotent element to a
regular unipotent element respects cohomological $A$-parameters. This is used
to give complete understanding of cohomological $A$-parameters for all
classical groups. We review the parametrization of Adams-Johnson packets of
cohomological representations of $G({\mathbb R})$ by cohomological
$A$-parameters and discuss various examples. We prove that the sum of the ranks
of cohomology groups in a packet on any real group (and with any infinitesimal
character) is independent of the packet under consideration, and can be
explicitly calculated. This result has a particularly nice form when summed
over all pure inner forms.
