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Moment-angle complexes are spaces acted on by a torus and parametrised by finite simplicial complexes.They are central objects in toric topology,and currently are gaining much interest in homotopy theory.Due the their combinatorial origins,moment-angle complexes also find applications in combinatorial geometry and commutative algebra.After an introductory part describing the general properties of moment-angle manifolds and complexes we shall concentrate on the complex-analytic aspects of the theory.Moment-angle manifolds provide a wide class of examples of non-Kaehler compact complex manifolds.A complex moment-angle manifold Z is constructed via a certain combinatorial data,called a complete simplicial fan.In the case of rational fans,the manifold Z is the total space of a holomorphic bundle over a toric variety with fibres compact complex tori.By studying the Borel spectral sequence of this holomorphic bundle,we calculate the Dolbeault cohomology and Hodge numbers of Z.In general,a complex moment-angle manifold Z is equipped with a canonical holomorphic foliation F and an algebraic torus action transitive in the transverse direction.Examples of moment-angle manifolds include the Hopf manifolds,Calabi-Eckmann manifolds,and their deformations.We construct transversely Kaehler metrics on moment-angle manifolds,under some restriction on the combinatorial data.We prove that all Kaehler submanifolds in such a moment-angle manifold lie in a compact complex torus contained in a fiber of the foliation F.For a generic moment-angle manifold in its combinatorial class,we prove that all its subvarieties are moment-angle manifolds of smaller dimension.This implies,in particular,that its algebraic dimension is zero.This is joint work with Yuri Ustinovsky and Misha Verbitsky.