Abstract. We introduce a notion of n-quasi-categories as fibrant objects of a model category structure on presheaves on Joyal's n-cell category Θn. Our definition comes from an idea of Cisinski and Joyal. However, we show that this idea has to be slightly modified to get a reasonable notion. We construct two Quillen equivalences between the model category of n-quasi-categories and the model category of Rezk Θn-spaces, showing that n-quasi-categories are a model for (∞, n)-categories. For n = 1, we recover the two Quillen equivalences defined by Joyal and Tierney between quasicategories and complete Segal spaces.

We present a slight variation on a notion of weak ∞-groupoid introduced by Grothendieck in Pursuing Stacks and we study the homotopy theory of these ∞-groupoids. We prove that the obvious definition for homotopy groups of Grothendieck ∞-groupoids does not depend on any choice. This allows us to give equivalent characterizations of weak equivalences of Grothendieck ∞-groupoids, generalizing a well-known result for strict ∞-groupoids. On the other hand, given a model category M in which every object is fibrant, we construct, following Grothendieck, a fundamental ∞-groupoid functor Π∞ from M to the category of Grothendieck ∞-groupoids. We show that if X is an object of M, then the homotopy groups of Π∞(X) and of X are canonically isomorphic. We deduce that the functor Π∞ respects weak equivalences.

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We prove that the folk model structure on strict ∞-categories transfers to the category of strict ∞-groupoids (and more generally to the category of strict (∞, n)-categories), and that the resulting model structure on strict ∞-groupoids coincides with the one defined by Brown and Golasiński via crossed complexes.