We introduce structured decompositions, category-theoretic structures which simultaneously generalize notions from graph theory (including treewidth, layered treewidth, co-treewidth, graph decomposition width, tree independence number, hypergraph treewidth and H-treewidth), geometric group theory (specifically Bass-Serre theory), and dynamical systems (e.g. hybrid dynamical systems). We define width functors, which provide a compositional way to analyze and relate different structural complexity measures, and establish a general duality between decompositions and completions of objects.