TITLE: Most significant results in motivic integration theory?
QUESTION [26 upvotes]: I am thinking about reading a course on motivic integration. I have already read certain introductions to the subject; yet I am not sure that they mention all the significant parts of the theory. So, my questions are:

Which papers and results are the most important for the theory of motivic integrations?
What are the main applications of motivic integration?
Would it be wise to avoid all quantifier elimination and related things in the course?

REPLY [3 votes]: Other very significant applications have to do with 'transfer theorems' of statements between local fields of positive and zero characteristic, in particular a different proof that the "fundamental lemma", as proved by Ngo in positive characteristic, is valid in characteristic zero: the first proof is by Waldspurger, but it can be proved using the (exponential) motivic integration theory of Cluckers and Loeser (see a survey of Cluckers, Hales and Loeser, e.g. on Cluckers's homepage; the foundational papers of Cluckers and Loeser are also there).  
Even more recent applications of these ideas (due in particular to Cluckers, Gordon and Halupczok) involve transfering integrability statements and have applications to harmonic analysis over local fields. (Interestingly, in some cases, the transfer goes in the other direction here: from known results in characteristic zero to results in positive characteristic).  
These results use a fair amount of model theory (in particular things known as the Denef-Pas language, and Presburger arithmetic).