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.