I will discuss the cohomology with coefficients in rank one local systems for various polyhedral products, including real Davis-Januszkiewicz spaces and toric complexes. As one application (joint work with Alvise Trevisan), I will show how to determine the Betti numbers and the cup products of real, quasi-toric manifolds. As another application (joint work with Graham Denham and Sergey Yuzvinsky), I will explain why Cohen-Macaulay toric complexes enjoy a certain "abelian duality" property.