The "polyhedral product functor" produces a space from a simplicial complex L
and a collection of pairs of spaces, {(A(i),B(i))}, where i ranges over the
vertex set of L. We give necessary and sufficient conditions for the resulting
space to be aspherical. There are two similar constructions, each of which
starts with a space X and a collection of subspaces, {X_i} and then produces a
new space. We give conditions for the results of these constructions to be
aspherical. All three techniques can be used to produce examples of closed
aspherical manifolds.