  A stratified space is a topological space together with a decomposition into
strata corresponding to different types of singularities. Examples of such
spaces appear everywhere in topology and geometry. The study of stratified
spaces involves invariants such as intersection cohomology which are only
invariants under stratification-preserving homotopies. In this thesis, we study
the homotopy theory of stratified spaces with respect to those stratified
homotopies. To do so, we construct model categories for stratified spaces and
we introduce new invariants to characterize them, the filtered homotopy groups.
A stratified space can be seen as a topological space $X$ together with a
continuous map to a poset of strata $X\to P$. We begin our study by restricting
ourselves to the filtered case, where the poset of strata is fixed. We define
the model category of filtered simplicial sets, show that it admits a
description "\`a la Kan", and characterize its weak-equivalences using the
filtered homotopy groups. We deduce a proof of a filtered version of Whitehead
theorem. We then construct a model category of filtered spaces. Its
weak-equivalences are the morphisms that induce isomorphisms on all filtered
homotopy groups, and its fibrations satisfy a filtered version of Serre's
lifting conditions. We show that it is Quillen-equivalent to a category of
diagrams of simplicial sets. We then work toward a comparison between the model
categories of filtered simplicial sets and of filtered spaces. They are
connected by a Quillen-adjunction, similar to the classical Kan-Quillen
adjunction. We conjecture that it is in fact a Quillen equivalence. Lastly,
working with the notion of Quillen bifibration, we show that there are model
categories of stratified spaces and of stratified simplicial sets. We show that
the two are related by an adjunction that preserve weak equivalences.
