  We review the duality between heterotic and F--theory string vacua with N=1
space-time supersymmetry in eight, six and four dimensions. In particular, we
discuss two chains of four-dimensional F--theory/heterotic dual string pairs,
where F--theory is compactified on certain elliptic Calabi-Yau fourfolds, and
the dual heterotic vacua are given by compactifications on elliptic Calabi-Yau
threefolds plus the specification of the $E_8\times E_8$ gauge bundles. We show
that the massless spectra of the dual pairs agree by using, for one chain of
models, an index formula to count the heterotic bundle moduli and determine the
dual F--theory spectra from the Hodge numbers of the fourfolds and of the type
IIB base spaces. Moreover as a further check, we demonstrate that for one
particular heterotic/F--theory dual pair the N=1 superpotentials are the same.
