  We describe the first order moduli space of heterotic string theory
compactifications which preserve $N=1$ supersymmetry in four dimensions, that
is, the infinitesimal parameter space of the Strominger system. We establish
that if we promote a connection on $TX$ to a field, the moduli space
corresponds to deformations of a holomorphic structure $\bar D$ on a bundle
$\cal Q$. The bundle $\cal Q$ is constructed as an extension by the cotangent
bundle $T^*X$ of the bundle $E= {\rm End}(V) \oplus {\rm End}(TX) \oplus TX$
with an extension class $\cal H$ which precisely enforces the anomaly
cancelation condition. The deformations corresponding to the bundle $E$ are
simultaneous deformations of the holomorphic structures on the poly-stable
bundles $V$ and $TX$ together with those of the complex structure of $X$. We
discuss the fact that the "moduli" corresponding to ${\rm End}(TX)$ cannot be
physical, but are however needed in our mathematical structure to be able to
enforce the anomaly cancelation condition. In the Appendix we comment on the
choice of connection on $TX$ which has caused some confusion in the community
before. It has been shown by Ivanov and others that this connection should also
satisfy the instanton equations, and we give another proof of this fact.
