In Section 2 we discuss motivations from the Conformal Field Theory. In Section 3 we formulate the conjectures about analytic and geometric pictures of the large complex structure limit. In Section 4 we describe a general framework of $A_{\infty}$-pre-categories adapted to the transversality problem in the definition of the Fukaya category. Section 5 is devoted to the Fukaya category and its degeneration. The reader will notice an advantage of working over the field of Laurent power series: one can consider all local systems over Lagrangian submanifolds, while in the conventional approach unitarity of the holonomy is required. Section 6 is devoted to the $A_{\infty}$-category of smooth functions introduced by Fukaya (and then studied by Fukaya and Oh in [FuO]). We prove that this $A_{\infty}$-category has very simple de Rham model. This part of the paper can be read independently of the rest. On the other hand, the technique of the proof will be used later in the paper. One important technical tool is an explicit $A_{\infty}$-structure on a subcomplex of a differential-graded algebra (see [GS], [Me]). We restate the formulas from [Me] in term of sums over a set of planar trees. The proof of the equivalence of Morse and de Rham $A_{\infty}$-categories uses the technique of [HL]. Section 7 is devoted to the analytic side of the homological mirror conjecture. We give a construction of mirror symmetry functor for torus fibrations in terms of the non-archimedean geometry. The use of non-archimedean analysis allows us to avoid problems with convergence of series in the definition of the Fukaya category. In Section 8 we construct an $A_{\infty}$-pre-category which is equivalent to a full $A_{\infty}$-subcategory of the derived category coherent sheaves on the Calabi-Yau manifold over ${{\bf C}}_{\varepsilon}$. Similarly to the comparison of Morse and de Rham pictures, we will prove that this category is also equivalent to an $A_{\infty}$-subcategory of the Fukaya category of the mirror dual torus fibration. In Appendix (Section 9) we describe the analogs of our constructions in the case of complex geometry.