This is more a comment than an answer, but its length makes me post it as an answer. I want to react to what I have just read, for the first time, about "Pursuing Stacks" at the nLab, and the words used there as well as in your question. I find it extremely irritating when people only use words such as "ideas" and "conjectures" to describe the content of "Pursuing Stacks". It makes me wonder whether they have read it or not. The words "rigorous", "results" or "theorems" are not used unless they describe other people's work. This is perfectly true that you would find ideas and conjectures in "Pursuing Stacks", but it does not prevent many notions and *results* regarding them to be "rigorously worked". It absolutely blows my mind that, nearly thirty years after the writing of this text, people still talk about it in that disrespectful way. (Those who do not think this is disrespectful to credit Grothendieck with "ideas" and "conjectures", no matter how "deep" or "beautiful", while others get the credit for the rigorous results, can read "Récoltes et semailles", where this question is precisely addressed at length.)

This is nevertheless true that "Pursuing Stacks" contain many ideas and conjectures, and David Roberts's answer gives, to my very partial knowledge, an accurate rough overview of what has been developed since then. I would just mention the fact that Grothendieck did not view simplicial sets as more homotopically relevant than other test categories. As regards $\infty$-groupoids, his approach is purely algebraic. Therefore, while I certainly do not claim that one of the approaches is better than the others, the prevalent simplicial approaches do not seem to me to be the one advocated by Grothendieck.

EDIT 1 (10 December 2012): This "answer" has got one downvote and someone has canceled their upvote. Perhaps I should have explained more. I find the question and the answers given so far quite sad because they all point out the fact that almost nobody has read "Pursuing Stacks", and people mostly talk about it by word of mouth. If some people had read it carefully, then sure enough they would have cited accurately some of Grothendieck's results. What are these precises references in the literature? I only know that there was a working group organized by Bénabou in the eighties, devoted to Pursuing Stacks, soon after its sending, but it quickly came to an end, and that someone in the Netherlands asked a student, whose name nobody seems to recall, to work on test categories, which he did not pursue very far. There certainly has been a lot of work in "categorical homotopy theory" since the writing of "Pursuing Stacks", but have Lurie, Rezk, Toën, Vezzosi (all cited by David Roberts) read "Pursuing Stacks" (I certainly do not claim they should have)? I think one should be very careful before claiming that one's own work or other people's work is indeed related to the content of "Pursuing Stacks" and even more before stating that it "realize" (?) or "formalize" some "ideas" of "Pursuing Stacks". To some extent, this text is like the Bible: eveybody talks about it but almost nobody knows what is written there.