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The moduli space of stable curves of Deligne and Mumford is acompactification of the moduli space of smooth curves of genus >=2 thatparametrizes certain nodal curves. It is a powerful tool for the study ofalgebraic curves. Higher-dimensional analogues were constructed by Koll\'ar,Shepherd-Barron and Alexeev in dimension 2, and by Viehweg in the case ofsmooth varieties. We will explain the recent ideas allowing for theconstruction of these moduli spaces in general, including the stable reductiontheorem in higher dimension, which reflects their compactness.  L'espace de modules des courbes stables de Deligne et Mumford est unecompactification de l'espace de modules des courbes lisses de genre >=2,param\'etrant certaines courbes nodales. C'est un outil puissant pour l'\'etudedes courbes alg\'ebriques. Des analogues en dimension sup\'erieure ont \'et\'econstruits par Koll\'ar, Shepherd-Barron et Alexeev en dimension 2, et parViehweg dans le cas des vari\'et\'es lisses. Nous expliquerons les id\'eesr\'ecentes ayant permis la construction de ces espaces de modules eng\'en\'eral, notamment le th\'eor\`eme de r\'eduction stable en dimensionsup\'erieure, qui refl\`ete leur compacit\'e.

R\'eduction stable en dimension sup\'erieure [d'apr\`es Koll\'ar, Hacon-Xu...]