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](https://www.google.com.au/search?q=letter+to+breen+grothendieck) 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](http://www.math.jussieu.fr/~maltsin/) has extracted [Grothendieck's proposed definition](http://www.math.jussieu.fr/~maltsin/ps/infgrart.pdf) for a weak $\infty$-groupoid, and there is work by [Ara](http://www.normalesup.org/~ara/recherche.html.en) towards showing that this definition satisfies the homotopy hypothesis.

The other parts (not entirely inseparable) are the first thoughts on [derivators](http://ncatlab.org/nlab/show/derivator), which were later taken up in great detail in Grothendieck's 1990-91 [notes](http://www.math.jussieu.fr/~maltsin/groth/Derivateursengl.html) (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](http://www.math.univ-montp2.fr/~toen/), [Vezzosi](http://www.dma.unifi.it/~vezzosi/) and others on _homotopical algebraic geometry_ (e.g. [HAG I](http://www.dma.unifi.it/~vezzosi/papers/hagI.pdf), [HAG II](http://www.dma.unifi.it/~vezzosi/papers/hagIIfin.pdf)) using simplicial sheaves on schemes. This has taken off with work of [Lurie](http://www.math.harvard.edu/~lurie/), [Rezk](http://www.math.uiuc.edu/~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](http://ncatlab.org/nlab/show/model+structure+on+simplicial+sets#joyals_model_structure_37) on the category of simplicial sets to give a basis to some of the ideas being tossed around at the time.