We start from a simplicial complex K on the set of m elements. We additionally, assume K to be a sphere triangulation given by some complete simplicial fan.

Following an exposition by Panov and Ustinovsky, we will construct smooth structures on moment-angle complex ZK for such a simplicial complex K. We do that by introducing an action of a real analytical subgroup of the complex torus. Our approach bears a striking resemblance to the Batyrev--Cox quotient construction for toric varieties. This fact becomes even more evident when we move to the world of complex manifolds. It turns out that even-dimensional moment-angle manifolds could be endowed with a structure of a complex quotient manifold.

Moment-angle manifolds provide a rich family of examples of non-Kähler manifolds with interesting geometry. The family of Hopf manifolds, for instance, is just a particular case of moment-angle manifolds. For the special case, when the data defining the fan is rational, moment-angle manifold attains a structure of principal bundle over the toric variety given by the same fan. This fact gives a fruitful link to the well-understood world of toric geometry. The existence of such a bundle allows one to compute various invariants of moment-angle manifolds, such as Dolbeault cohomology and automorphism groups.