  The non-perturbative superpotential can be effectively calculated in
$M$-theory compactification to three dimensions on a Calabi-Yau four-fold $X$.
For certain $X$, the superpotential is identically zero, while for other $X$, a
non-perturbative superpotential is generated. Using $F$-theory, these results
carry over to certain Type IIB and heterotic string compactifications to four
dimensions with $N=1$ supersymmetry. In the heterotic string case, the
non-perturbative superpotential can be interpreted as coming from space-time
and world-sheet instantons; in many simple cases contributions come only from
finitely many values of the instanton numbers.
