  For each $n \geq -1$, a quasi-category is said to be $n$-truncated if its
hom-spaces are $(n-1)$-types. In this paper we study the model structure for
$n$-truncated quasi-categories, which we prove can be constructed as the
Bousfield localisation of Joyal's model structure for quasi-categories with
respect to the boundary inclusion of the $(n+2)$-simplex. Furthermore, we prove
the expected Quillen equivalences between categories and $1$-truncated
quasi-categories and between $n$-truncated quasi-categories and Rezk's
$(n,1)$-$\Theta$-spaces.
