The nerve functor from the category of small categories to the category of simplicial sets is full and faithful, and the image is described by a sheaf condition. Furthermore, the category of small categories has a closed model structure in which a functor is a weak equivalence if and only if the nerve is a weak equivalence of simplicial sets, and there is a Quillen equivalence between the categories of small categories and simplicial sets. This paper addresses the same issues for higher categories. An \(\omega\)-graph (also known as a globular set) is the higher-dimensional analog of a graph; it is a diagram of sets and ``source/target operations'' \[ \dots\rightrightarrows X_n\rightrightarrows\dots\rightrightarrows X_1\rightrightarrows X_0 \] satisfying the necessary relations. An \(\omega\)-category is an \(\omega\)-graph equipped with composition laws and identities satisfying the analogs of Godement's interchange rules. In the language of \textit{M. A. Batanin} [Adv. Math. 136, 39-103 (1998; Zbl 0912.18006)], a weak \(\omega\)-category is an algebra over a contractible \(\omega\)-operad. The \(\omega\)-categories are then the algebras over the terminal \(\omega\)-operad coming from the free/forgetful pair between \(\omega\)-categories and \(\omega\)-graphs. Given an \(\omega\)-operad \(A\), the author defines a nerve functor. Just as for the usual case, the nerve is given by a dense subcategory \(\Theta_A\) of the category of \(A\)-algebras, and is a fully faithful functor from \(A\)-algebras to presheaves on \(\Theta_A\). The image is characterized by a condition analogous to that of the usual nerve. In the terminal case, the author proves that \(\Theta_A\) coincides with Joyal's cell category \(\Theta\). Hence \(\Theta\) plays the same role for weak \(\omega\)-categories as Segal's \(\Gamma\) plays for \(E_\infty\)-spaces. The author proceeds by defining closed model structures on cellular sets (i.e., presheaves on \(\Theta\)) and cellular spaces, which are Quillen equivalent to the usual one on topological spaces. For an \(\omega\)-operad \(A\), the canonical map \(\Theta_A\to \Theta\) gives a closed model structure on the category of \(A\)-cellular spaces (simplicial presheaves on \(\Theta_A\)). The author then uses the nerve to get a closed model structure on the category of simplicial \(A\)-algebras which is Quillen equivalent to the category of \(A\)-cellular spaces. If \(A\) is a contractible \(\omega\)-operad this gives a Quillen equivalence between the closed model structure on simplicial \(A\)-algebras and cellular spaces. Furthermore, both the homotopy categories in question are spanned by simplicially discrete objects, so the associated homotopy categories of \(A\)-algebras and cellular sets are equivalent. The paper ends with a list of open questions.