  We show that the classical equations of motion of the low-energy effective
field theory describing the massless modes of the heterotic (or type I) string
admit two classes of supersymmetric self--dual backgrounds. The first class,
which was already considered in the literature, consists of solutions with a
(conformally) flat metric coupled to axionic instantons. The second includes
Asymptotically Locally Euclidean (ALE) gravitational instantonic backgrounds
coupled to gauge instantons through the so--called ``standard embedding''. We
show that some elements of these two classes of solutions are dual to each
other in the sense of Buscher's duality. We give a world--sheet interpretation
of the heterotic ALE istanton solutions in terms of superconformal $N=(4,4)$
$\s$--models and argue for their validity to all orders in $\alpha^\prime$.
Specializing the gravitational background to the Eguchi--Hanson instanton, we
compute the indices of the fermionic operators and give the explicit form of
all the relevant fermionic and bosonic zero--modes.}
