  We present a method for explicitly computing the non-perturbative
superpotentials associated with the vector bundle moduli in heterotic
superstrings and M-theory. This method is applicable to any stable, holomorphic
vector bundle over an elliptically fibered Calabi-Yau threefold. For
specificity, the vector bundle moduli superpotential, for a vector bundle with
structure group G=SU(3), generated by a heterotic superstring wrapped once over
an isolated curve in a Calabi-Yau threefold with base B=F1, is explicitly
calculated. Its locus of critical points is discussed. Superpotentials of
vector bundle moduli potentially have important implications for small
instanton phase transitions and the vacuum stability and cosmology of
superstrings and M-theory.
