  We continue the study of the Strominger-Yau-Zaslow mirror symmetry
conjecture. Roughly put, this states that if two Calabi-Yau manifolds X and Y
are mirror partners, then X and Y have special Lagrangian torus fibrations
which are dual to each other. Much work on this conjecture is necessarily of a
speculative nature, as in dimension 3 it is still a very difficult problem of
how to construct such fibrations. Nevertheless, assuming the existence of such
fibrations there are many things one can prove. This paper covers a number of
issues. First it applies results from the theory of completely integrable
hamiltonian systems to understand some aspects of the geometry of such
fibrations. From this, using reasonable regularity assumptions on the
fibrations, one can understand how the cohomology of dual fibrations are
related. We then study the question of how, given one such fibration, one would
put a symplectic and complex structure on the dual fibrations, generalising
work of Hitchin. While this question cannot be answered at this stage, these
results should give insight into the nature of the problem. We sum up these
ideas in a refined version of the Strominger-Yau-Zaslow conjecture.
  Finally, to give evidence for this conjecture, we prove it explicitly for K3
surfaces. One finds a construction of mirror symmetry for K3 surfaces which
does not require the use of Torelli theorems, and is much more differential
geometric in nature than previous constructions.
