The cosmopolitan relevance of partially ordered mathematical structures in chemistry is argued. Many examples are briefly noted, including those involving chemical periodicities, reactivities, aromaticities, electronegativities, molecular branching, molecular shapes, symmetries, complexities, curve fittings, and more. A few fundamental theorems concerning metrics (or distance functions) on partially ordered sets are noted, first for the intuitively appealing ''scaled'' posets, then for the more general ''transformed'' posets. Interspersed along the way are a few examples which are developed to a greater extent, including Randic-Wilkins periodicity for alkanes ; the general concept of aromaticity ; molecular branching ; least-squares fittings ; and (most extensively) molecular shapes, chiralities, and symmetries. In each of these types of examples clarifications, alternative views, and extensions of previous works result.