Do you understand SYZ conjecture The aim of this question is to understand SYZ conjecture ("Mirror symmetry is T-Duality").
I don't expect a full and quick answer but to find a better picture from answers and comments.
The whole idea is to construct the Mirror C.Y $Y$ from $X$ intrinsically as follows.
One considers the moduli of special Lagrangian tori with a flat $U(1)$ bundle on it in $X$.
Then we put a metric on this moduli (plus corrections coming from J-holomorphic disks) and expect that this moduli and the metric given on it is the mirror C.Y we were looking for.
Here are the things I can not understand:
---What is the metric given in the paper "Mirror symmetry is T-Duality"?
Where does it come from? (I can not understand the formulation of metric there).
and more importantly
--How do we deform the metric using J-holomorphic disks(instantons)?
 A: Here are some papers on SYZ worth reading:


*

*Hitchin's "The moduli space of special Lagrangian submanifolds"  arXiv:dg-ga/9711002

*M. Gross's  survey
Hitchin's paper was written shorly after Mirror Symmetry is T-duality and it is a matematical 
explanation of the paper. Essentially Maclean proved that the moduli space of sL submanifolds is unobstructed and its tangent space is the space of harmonic 1-forms on the sL submanifold. A natural metric which you can put on the moduli space is the $L^2$ metric on harmonic forms. When the sL submanifold is a torus, the moduli space also has an "affine structure". This was already known from integrable systems, they are called action coordinates. They are affine because they are defined up to affine transformations (with linear part having integral coefficients). Hitchin shows that with respect to these coordinates the metric can be expressed as the Hessian of a function. Hitchin also shows that the moduli space has two affine strutures (this is because of the "special" condition). The two affine structures are related by Legendre transform using the Hessian (i.e.the metric). So one could say that mirror symmetry is "Legendre transform". 
Things have developed a lot since Hitchin's paper, and M. Gross surveys these developements. How to do "quantum corrections" to the metric is a major open problem, there are many approaches. They seem all quite difficult.... Auroux in the paper mentioned above deals with it. I heard a talk of Fukaya where he wants do do it with Floer homology for families, but I do not know much about it. Then there is the program of Kontsevich and Soibelmann, using rigid analytic geometry and the Gross-Siebert program. It seems that quantum corrections could be understood in terms of "tropical geometry" in the moduli space of SL tori (an "affine manifold with sigularities"). In a recent paper of M. Gross ("Mirror symmetry for $\mathbb{P}^{2}$ and tropical geometry"), he explains how "period calculations" can be understood in terms of tropical geometry (at least for $\mathbb{P}^{2}$).  Here  you can find a link to a book of M. Gross where he explains the connection between tropical geometry and mirror symmetry. 
A: You might also want to check out 
Riemannian Holonomy Groups and Calibrated Geometry by Dominic Joyce.
In Chapter 9, he gives a nice introduction to SYZ that is very accessible. He also points out reasons why SYZ as originally formulated (even though it does not have a precise formulation) could not be true. He proposes modified versions of SYZ that he believes are likely to be closer to any eventual true statement. (I have not looked at Auroux's recent work, so I can't comment on that.)
A: Hi-
Just saw this thread.  Maybe I should comment.  The conjecture
can be viewed from the perspective of various categories:
geometric, symplectic, topological.  Since the argument is
physical, it was written in the most structured (geometric)
context -- but it has realizations in the other categories
too.
Geometric:  this is the most difficult and vague, mathematically,
since the geometric counterpart of even a conformal field theory
is approximate in nature.  For example, a SUSY sigma model with
target a compact complex manifold X is believed to lie in the
universality class of a conformal field theory when X is CY,
but the CY metric does not give a conformal field theory on
the nose -- only to one loop.  Likewise, the arguments about
creating a boundary conformal field theory using minimal (CFT) +
Lagrangian (SUSY) are only valid to one loop, as well.
To understand how the corrections are organized, we should
compare to (closed) GW theory, where "corrections" to the classical
cohomology ring come from worldsheet instantons -- holomorphic
maps contributing to the computation by a weighting equal
to the exponentiated action (symplectic area).  The "count"
of such maps is equivalent by supersymmetry to an algebraic
problem.  No known quantity (either spacetime metric or
Kahler potential or aspect of the complex structure) is
so protected in the open case, with boundary.  That's why
the precise form of the instanton corrections is unknown,
and why traction in the geometric lines has been made
in cases "without corrections" (see the work of Leung, e.g.).
Nevertheless, the corrections should take the form of 
some instanton sum, with known weights.  The sums seem
to correspond to flow trees of Kontsevich-Soibelman/
Moore-Nietzke-Gaiotto/Gross-Siebert, but I'm already running
out of time.
Topological:  Mark Gross has proven that the dual torus
fibration compactifies to produce the mirror manifold.
Symplectic:  Wei Dong Ruan has several preprints which
address dual Lagrangian torus fibrations, which come
to the same conclusion as Gross (above).  I don't know
much more than that.
Also-
Auroux's treatment discusses the dual Lagrangian
torus fibration (even dual slag, properly understood)
for toric Fano manifolds, and produces the mirror
Landau-Ginzburg theory (with superpotential) from this.
With Fang-Liu-Treumann, we have used T-dual fibrations
for the same fibration to map holomorphic sheaves
to Lagrangian submanifolds, proving an equivariant version of
homological mirror symmetry for toric varieties.
(There are many other papers with similar results
by Seidel, Abouzaid, Ueda, Yamazaki, Bondal, Auroux,
Katzarkov, Orlov -- sorry for the biased view!)
Reversing the roles of A- and B-models, Chan-Leung
relate quantum cohomology of a toric Fano to the
Jacobian ring of the mirror superpotential via T-duality.
Help or hindrance?
