Mathematics of AdS/CFT To date, what is the most mathematically precise formulation of the AdS/CFT correspondence, and what are the most robust tests of the conjecture?
 A: when one talks about "truly rigorous" mathematical physics, there is really no good treatment of ordinary quantum field theory yet. So of course, there is no "truly rigorous" framework to discuss AdS/CFT whose one side is a quantum field theory and another side is something even more complicated - a description of a quantum gravity theory in terms of string/M-theory.
At the physics level, nothing has changed since 1997 or early 1998: the claim was known in the right form from the very beginning. The string theory dynamics in the AdS space is exactly equivalent to the dynamics of the field theory on the boundary. While the field theory may be defined by the lattice - up to some issues with adjusting supersymmetry at long distances - the gravitating string theory side is only known from various limiting descriptions, including perturbative string theory, Matrix theory, and other dual descriptions, besides various terms calculable from SUSY etc. But there is no counterpart of the "lattice" that would allow us to define string theory "completely exactly" in any background.
Nathan Berkovits has made the longest steps to prove AdS/CFT in a Gopakumar-Vafa way, using his pure spinors etc: the world sheet of string theory directly degenerates into Feynman diagrams in the right limit. However, no physicist is too curious about such things because the map clearly works. I believe that the tests derived from the BMN-pp-wave duality - which show that even all strings and their interactions on the AdS side exactly match to the boundary CFT side - are among the most stringent tests of the AdS/CFT correspondence - especially for the canonical background AdS5 x S5 of type IIB related to N=4 SYM in d=4.
This industry has transformed to the integrability business in recent years and increasingly began to overlap with the research of the people who study N=4 SYM scattering amplitudes via twistors although the role of the AdS gravity in the latter remains largely invisible.
Best wishes
Lubos
A: I'm not sure what you have in mind with "mathematically precise"; as with almost anything involving quantum field theory or string theory, there's no rigorous definition of the theories involved in the duality. But, if you grant me their existence, I would say the sharpest statement is still the early formulation by Gubser, Klebanov, and Polyakov and by Witten, i.e. that the partition function of the CFT in the presence of external sources for single-trace operators is the same as that for string theory in AdS with boundary conditions determined by the sources.
The most detailed computational checks of the correspondence are probably those that use integrability to compute anomalous dimensions of operators over the full range from weak to strong coupling. I'm not an expert on this, but I'll point you to one fairly recent paper that contains some of the major references to get you started.
From a more global perspective, though, gauge/gravity duality extends well beyond the original case of ${\cal N}=4$ SYM and $AdS_5 \times S^5$, to any theory that meets the two requirements of having a large-$N$ expansion and a large 't Hooft coupling. The important ideas, again, were mostly there in the very early papers, but I would say they've been put on a somewhat more solid footing. The key point is that the bulk theory is tractable in the case when only a few fields are involved and curvatures are weak. This condition, translated to a statement about the dual field theory, is that most of the single-trace operators acquire very large anomalous dimensions (which is natural in very strongly coupled theories). Recently there has been some progress in formulating a bottom-up argument from the opposite direction, i.e. starting with the assumption that a large-$N$ conformal field theory includes few low-dimensional single-trace operators and arguing that this implies the existence of a bulk dual theory. See this paper of Heemskerk et al..
I don't know if any of this is what you would think of as "mathematics"....
A: The most robust experimental test of AdS/CFT so far is the measurement of the viscosity of the quark-gluon plasma formed in the aftermath of collisions between heavy atoms. There is a nice paper on arXiv about AdS-CFT and the RHIC fireball, where RHIC is the relativistic heavy ion collider at Brookhaven. And another nice review paper on the underlying physics by Son and Starinets. You might also find stimulating Subir Sachdev's work relating AdS/CFT to problems in condensed matter.
                              All the best,

