  In this paper we discuss two major conjectures in Mirror Symmetry:
Strominger-Yau-Zaslow conjecture about torus fibrations, and the homological
mirror conjecture (about an equivalence of the Fukaya category of a Calabi-Yau
manifold and the derived category of coherent sheaves on the dual Calabi-Yau
manifold). Our point of view on the origin of torus fibrations is based on the
standard differential-geometric picture of collapsing Riemannian manifolds as
well as analogous considerations for Conformal Field Theories. It seems to give
a description of mirror manifolds much more transparent than the one in terms
of D-branes.
  Also we make an attempt to prove the homological mirror conjecture using the
torus fibrations. In the case of abelian varieties, and for a large class of
Lagrangian submanifolds, we obtain an identification of Massey products on the
symplectic and holomorphic sides. Tools used in the proof are of a mixed
origin: not so classical Morse theory, homological perturbation theory and
non-archimedean analysis.
