  We attempt to settle the issue as to what is the correct non-abelian
generalisation of the Born-Infeld action, via a consideration of the two-loop
$\beta$--function for the non-abelian background gauge field in open string
theory. An analysis of the bosonic theory alone shows the recent proposal of
Tseytlin's to be somewhat lacking. For the superstring, however, this proposal
would seem to be correct, and not just within the approximation used in
\cite{tseytlin}. Since it is this latter case that is relevant to the
description of D-branes we, in effect, obtain an independent verification of
Tseytlin's result. Some issues involved in the concept of non-abelian
T--duality are discussed; and it is shown how the interaction between separated
and parallel branes, in the form of massive string states, emerges.
