  We describe the action of the mod $2$ Steenrod algebra on the cohomology of
various polyhedral products and related spaces. We carry 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, we deduce some consequences
for the lowest cohomological dimension in which non-trivial Steenrod operations
can appear.
  We present a version of cochain-level formulas for Steenrod operations on
simplicial complexes. We explain the idea of "propagating" such formulas from a
simplicial complex $K$ to polyhedral joins over $K$ and we give examples of
this process. We tie the propagation of the Steenrod algebra actions on
polyhedral joins to those on moment-angle complexes. Although these are cases
where one can understand the Steenrod action via a stable homotopy
decomposition, we anticipate applying this method to cases where there is no
such decomposition.
