I will describe the action of the mod 2 Steenrod algebra on the cohomology of various polyhedral products and related spaces. I’ll illustrate this out for Davis-Januszkiewicz spaces and their generalizations, for moment-angle complexes as well as for certain polyhedral joins. By studying the combinatorics of underlying simplicial complexes, some consequences will be deduced for the lowest cohomological dimension in which non-trivial Steenrod operations can appear.

In addition, I will explain the idea of "propagating'' the Steenrod algebra action from a simplicial complex K to polyhedral joins over K. Naturally, I will tie the propagation of the Steenrod algebra actions on polyhedral joins to those on moment-angle complexes.