  The moment-angle complex Z_K is cell complex with a torus action constructed
from a finite simplicial complex K. When this construction is applied to a
triangulated sphere K or, in particular, to the boundary of a simplicial
polytope, the result is a manifold. Moment-angle manifolds and complexes are
central objects in toric topology, and currently are gaining much interest in
homotopy theory, complex and symplectic geometry.
  The geometric aspects of the theory of moment-angle complexes are the main
theme of this survey. We review constructions of non-Kahler complex-analytic
structures on moment-angle manifolds corresponding to polytopes and complete
simplicial fans, and describe invariants of these structures, such as the Hodge
numbers and Dolbeault cohomology rings. Symplectic and Lagrangian aspects of
the theory are also of considerable interest. Moment-angle manifolds appear as
level sets for quadratic Hamiltonians of torus actions, and can be used to
construct new families of Hamiltonian-minimal Lagrangian submanifolds in a
complex space, complex projective space or toric varieties.
