In the definition of moment-angle manifold in Davis-Januszkiewicz's 1991 paper, the orbit space is a simple convex polytope $P$ which is contractible. We can replace $P$ by an arbitrary smooth nice manifold with corners $Q$ and define the generalized moment-angle manifold $\mathcal{Z}_Q$ in the similar way as usual moment-angle manifold (see https://arxiv.org/abs/2011.10366). Let $F_1,\cdots,F_m$ be all the facets of $Q$ and $\lambda: \{ F_1,\cdots,F_m \} \rightarrow
\mathbb{Z}^m$ be a map such that $\{\lambda(F_1),\cdots,\lambda(F_m)\}$ is a unimodular basis of $\mathbb{Z}^m\subset \mathbb{R}^m=T_e(S^1)^m$.
\begin{equation}
\mathcal{Z}_{Q} = Q\times (S^1)^m / \sim
\end{equation}
where $(x,g) \sim (x',g')$ if and only if $x=x'$ and $g^{-1}g' \in \mathbb{T}^{\lambda}_x$
where $\mathbb{T}^{\lambda}_x$ is the subtorus of $(S^1)^m$ determined by
the linear subspace of $\mathbb{R}^m$ spanned by the set $\{ \lambda(F_j) \, |\, x\in F_j \}$.
The free quotient of $\mathcal{Z}_Q$ under the action of some subtorus in $(S^1)^m$ of rank $m-n$ gives analogues of toric manifolds in this setting (where $n$ is the dimension of $Q$). Such a space can also be defined from a (non-degenerate) characteristic function on the facets of $Q$. The equivariant cohomology ring of $\mathcal{Z}_Q$ with $\mathbf{k}$-coefficients is isomorphic to a ring $\mathbf{k}\langle Q\rangle$ (called the topology face ring of $Q$) that is determined not only by the face poset of $Q$ but also by the $\mathbf{k}$-cohomology rings of all the faces of $Q$. The definition of topology face ring is a direct generalization of the face ring of a simple convex polytope.
Furthermore, we can replace $Q$ by an arbitrary finite CW-complex $X$ with a panel structure $\mathcal{P}$ and define moment-angle complex $(D^2,S^1)^{(X,\mathcal{P})}$ and do the similar calculations for its cohomology and equivariant cohomology (see https://arxiv.org/abs/2103.04281). The definition of panel structure
is due to M. Davis's 1983 paper "Groups generated by reflections and aspherical manifolds not covered by
Euclidean space" in Ann. of Math.
Roughly speaking, a panel structure on $X$ defines "abstract faces" on $X$ which allows us to do the similar construction as $\mathcal{Z}_Q$. But it is more convenient to think of
$(D^2,S^1)^{(X,\mathcal{P})}$ as the colimit of a diagram of CW-complexes of the form
$ f\times \underset{j\in I_f}{\prod} D^2_{(j)} \times
\underset{j\in [m]\backslash I_f}{\prod} S^1_{(j)}$ where $f$ ranges over all
the "abstract faces" of $(X,\mathcal{P})$ and $I_f$ denotes all indices of the panels that contain $f$. Note that "panels" plays the role of facets here.
In general, $(D^2,S^1)^{(X,\mathcal{P})}$ may not be a manifold. The free quotient of $(D^2,S^1)^{(X,\mathcal{P})}$ under some torus action can also be considered as a far-reaching generalization of manifolds with locally standard torus actions. In addition, the topology face ring of the panel structure $(X,\mathcal{P})$ also makes perfect sense, which is isomorphic to the equivariant cohomology ring of $(D^2,S^1)^{(X,\mathcal{P})}$.
In particular, when $X$ is the cone of the barycentric subdivision of a simplicial complex $K$ (with a canonical panel structure $\mathcal{P}_K$), the topological face ring of $(X,\mathcal{P}_K)$ is nothing but the face ring (Stanley-Reisner ring) of $K$ (see Section 5 of that paper).
