  We study aspects of Calabi-Yau four-folds as compactification manifolds of
F-theory, using mirror symmetry of toric hypersurfaces. Correlation functions
of the topological field theory are determined directly in terms of a natural
ring structure of divisors and the period integrals, and subsequently used to
extract invariants of moduli spaces of rational curves subject to certain
conditions. We then turn to the discussion of physical properties of the
space-time theories, for a number of examples which are dual to $E_8\times E_8$
heterotic N=1 theories. Non-critical strings of various kinds, with low tension
for special values of the moduli, lead to interesting physical effects. We give
a complete classification of those divisors in toric manifolds that contribute
to the non-perturbative four-dimensional superpotential; the physical
singularities associated to it are related to the apppearance of tensionless
strings. In some cases non-perturbative effects generate an everywhere non-zero
quantum tension leading to a combination of a conventional field theory with
light strings hiding at a low energy scale related to supersymmetry breaking.
