We study the sets of periods of triangular maps on a cartesian product of arbitrary spaces. As a consequence we extend Kloeden's Theorem (in a 1979 paper) to a class of triangular maps on cartesian products of intervals and circles. We also show that, in some sense, this is the more general situation in which the Sharkovskiĭ ordering gives the periodic structure of triangular maps. © 1993, Australian Mathematical Society. All rights reserved.