What do orbital integrals have to do with reciprocity? Hi, this is my first question (of many). I am blogging for the Fields Medal Symposium and would like to get into the mathematics involved with our program. 
In an attempt to sort through the articles and through all the conversations that I've been having in and around the Fields Institute, I'm having trouble seeing how all these concepts relate to each other. 
In particular, the Langlands Program began with an aim to discover quadratic reciprocity laws, for which methods of representation theory of automorphic forms were applied. How and when do orbital integrals come into the picture and why is stability important?
 A: One of the main problems in the Langlands program is functoriality, which is roughly speaking: A "map", say $G \mapsto G'$, between groups should "transfer" certain representations.
I will remain pretty vague what "map" and "transfer" are, simply because I only understand special cases of this. The representations are (cuspidal) automorphic representations. Essentially you want to show that automorphic forms are a category with the transforms being the morphisms.
I will describe the Jacquet-Langlands correspondence. A classical way to proof such transfers is via the trace formula. Roughly speaking, it is an identity between the representations and its conjugacy classes. It suffices for proving the transfer that the terms involving the conjugacy classes of $G$ and $G'$ coincide in some sense. Now, these terms are (weighted) orbital integrals associated to each conjugacy class.
Another part of Langlands program is the correspondence with Galois representations, which is probably what you mean with quadratic reciprocity in this context. Here, the strategy is similar only that given a Galois representation, you have to let it act on some geometry which is comparable with the conjugacy classes of $G$. I think that was the key ingredient in Laurent Lafforgue's proof for $GL(n)$ over a global function field.
So the trace formula is the standard strategy for both correspondence and functoriality. It is useful to have a good understanding of orbital integrals therefore.
Now let me address stability: Quote from Harris http://www.institut.math.jussieu.fr/projets/fa/bpFiles/Intro_Harris.pdf, Section 3, which is by the way an excellent survey.

Semi-simple conjugacy classes
  in inner forms of
  $GL(n)$ are completely characterized by their characteristic polynomials. Similar characterizations are possible for other
  groups, but only over an
  algebraically closed field.

Now, global fields are never algebraically closed and to overcome the issues involved in that matter is precisely the problem of stability, I think.
