An integral affine structure on a manifold of dimension $n$ is given by a torsion-free flat connection with the monodromy reduced to $GL(n,{{\bf Z}})$. There are two basic situations in which integral affine structures occur naturally. One is the case of classical integrable systems described briefly in Section 3. Most interesting for us is a class of examples arising from analytic manifolds over non-archimedean fields which is discussed in Section 4. It is motivated by the approach to Mirror Symmetry suggested in [KoSo]. We recall it in Section 5. From our point of view manifolds with integral affine structure appear in Mirror Symmetry in two ways. One considers the Gromov-Hausdorff collapse of degenerating families of Calabi-Yau manifolds. The limiting space can be interpreted either as a contraction (see Section 4.1) of an analytic manifold over a non-archimedean field of Laurent series ${\bf C}((t))$, or as a base of a fibration of a Calabi-Yau manifold by Lagrangian tori (with respect to the symplectic Kähler 2-form). On a dense open subset of the limiting space one gets two integral affine structures associated with two interpretations, the non-archimedean one and the symplectic one. Mirror dual family of degenerating Calabi-Yau manifolds should have metrically the same Gromov-Hausdorff limit, with the roles of two integral affine structures interchanged.

Very interesting question arises: how to reconstruct these families of Calabi-Yau manifolds from the corresponding manifolds with integral affine structures? This question was one of the main motivations for present work.