Conjectures in Grothendieck's "Pursuing stacks"
To dash off a quick answer, Pursuing Stacks is composed of (if memory serves correctly) three themes. The first was homotopy types as higher (non-strict) groupoids. This part was first considered in Grothendieck's letters to Larry Breen from 1975, and is mostly contained in the letter to Quillen which makes up the first part of PS (about 12 pages or so). Maltsiniotis has extracted Grothendieck's proposed definition for a weak $\infty$-groupoid, and there is work by Ara towards showing that this definition satisfies the homotopy hypothesis.
The other parts (not entirely inseparable) are the first thoughts on derivators, which were later taken up in great detail in Grothendieck's 1990-91 notes (see there for extensive literature relating to derivators, the first 15 of 19 chapters of Les Dérivateurs are themselves available), and the 'schematisation of homotopy types', which is covered by work of Toën, Vezzosi and others on homotopical algebraic geometry (e.g. HAG I, HAG II) using simplicial sheaves on schemes. This has taken off with work of Lurie, Rezk and others dealing with derived algebraic geometry, which is going far ahead of what I believe Grothendieck envisaged.
During correspondence with Grothendieck in the 80s, Joyal constructed what we now call the Joyal model structure on the category of simplicial sets to give a basis to some of the ideas being tossed around at the time.
Edit: I forgot something that is in PS, and that is the theory of localisers and modelisers, Grothendieck's conception of homotopy theory which you mention, which is covered in the work of Cisinski.
Edit 2019: Toën has a new preprint out
Bertrand Toën, Le problème de la schématisation de Grothendieck revisité, arXiv:1911.05509
with abstract starting
"The objective of this work is to reconsider the schematization problem of [Pursuing Stacks], with a particular focus on the global case over Z. For this, we prove the conjecture [Conj. 2.3.6 of Toën's Champs affines]..."
I agree with Tim that calling Pursuing Stacks a "letter to Quillen" is erroneous, especially as Quillen never replied. Grothendieck also wrote: "This is written in English in response to a correspondence in English." At one stage he planned more volumes in French but it seems got diverted from this. I hope the following will be of help, in addition to David Robert's answer, to put the situation with regard to models for homotopy theory in context.
In a letter dated 02/05/1983 Alexander Grothendieck wrote to me: "Don't be surprised by my supposed efficiency in digging out the right kind of notions--I have just been following, rather let myself be pulled ahead, by that very strong thread (roughly: understand non commutative cohomology of topoi!) which I kept trying to sell for about ten or twenty years now, without anyone ready to ``buy'' it, namely to do the work. So finally I got mad and decided to work out at least an outline by myself."
But this question is about the homotopy theory of categories and the related question of "Why simplicial sets?", for which see also the good answers in March 2011 to Is there a high-concept explanation for why "simplicial" leads to "homotopy-theoretic"?.
Dan Kan's first contribution to combinatorial homotopy was in terms of cubical sets. When he went to Princeton the disadvantages of cubical sets were found: cubical groups did not satisfy the extension property, and the geometric realisation of the cartesian product of cubical sets had the wrong homotopy type, while as shown by Moore and Milnor respectively, the situation was fine for simplicial sets. So they did not attempt to refine the cubical theory. In my doctoral studies at Oxford, 1956-59, the exposition was all simplicial, especially when Michael Barratt came back from Princeton in 1957.
However the 1960 book by Hilton and Wylie on algebraic topology was cubical, as were some 1962 notes of Federer from Brown University, a later book by Massey, and cubical sets continued to be found useful in various places. Our 2011 book on "Nonabelian algebraic topology" is almost entirely cubical, because of its emphasis on the use of higher homotopy Seifert-van Kampen theorems.
My 1965 intuition for using cubical sets was based on generalising the van Kampen theorem to higher dimensions. It seemed entirely reasonable that the above diagram could be expressed as: the big square is the composition of the little squares. C. Ehresmann's 1965 book on "Categories structuree" gave a definition of double categories which expressed this nicely. Indeed, I have answered this on mathoverflow as using matrix notation where $(a_{ij})$ denotes a composable array and $[a_{ij}]$ denotes the composite. So one has an easy definition of the $n$-fold cubes of the nerve of an $n$-fold category, except that there seems currently no name for the cubical type geometry underlying an $n$-fold category. Note that composable sequences of morphisms in a category are used in describing the nerve of a category, but it seems more difficult, at least for me, to define multiple compositions in simplicial or globular terms, although the simplicial nerve of an $n$-fold category is easily defined as an $n$-fold simplicial set.
By contrast, the singular cubical complex of a space, or filtered space, is ideally suited for the description of multiple compositions, using an array notation. I have already explained this in answer to this mathoverflow question.
So in considering what category in which to work the question of "what should be adequacy and convenience?" is crucial. It is as reasonable to ask this for combinatorial models of homotopy theory as it was in 1963 to ask it for categories for topology in my paper Ten topologies.
A property that was also required for the conjectured proof of a putative higher van Kampen theorem using homotopy classes of maps was the notion of "commuting cube", and that "any composition of commuting cubes is commutative". Chris Spencer and I found that the notion of "connection" for a double groupoid was good for this, and that it allowed an equivalence between crossed modules and single pointed edge symmetric double groupoids with connections. Then Philip Higgins and I found in 1974 the construction of the homotopy double groupoid of a pair $(X,A,x)$ of pointed spaces using homotopy classes of maps $I^2 \to X$ which take the edges to $A$ and the vertices to $x$. This gave the first homotopy fundamental double groupoid, which enabled the proof of a 2-dimensional van Kampen theorem, including the usual theorem for the fundamental group as a special case, not just an implication.
There are grounds for suggesting that simplicial sets are convenient, but are not entirely adequate, since they cannot easily express multiple compositions. On the other hand, cubical sets with connections are adequate for this test, but not entirely convenient! Andy Tonks proved that cubical groups with connection are Kan complexes. One reason for inconvenience is that although they have been shown to form a strict test category in the sense of Grothendieck, in the paper given here, the geometric realisation of the categorical product is only of the homotopy type of the product of the realisations, not actually homeomorphic to the product as in the case of simplicial sets. The latter homeomorphism property implies that in the right convenient category, the geometric realisation of a simplicial group is a topological group.
The cubical setup is also not sufficient to describe the geometry underlying $n$-fold categories, and indeed there seems currently no name for such a structure in which cubes have different types of faces in different directions. Yet Grothendieck remarked to me on my affirming Loday's theorem, that (strict) $n$-fold groupoids model weak homotopy $n$-types: "That is absolutely beautiful!"
So it as well not to assume we have the final story, and to investigate options!
January, 2015: This answer is related to my answer to
https://math.stackexchange.com/questions/1112107/why-does-seifert-van-kampen-not-hold-with-n-th-homotopy-groups/
November, 2016 There is more discussion in this preprint Modelling and Computing Homotopy Types: I.
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.