Constructive Neutral Evolution theory provides selectively neutral explanations of the origin and maintenance of biological complexity. This essay provides an analysis of CNE as an explanatory strategy defined by a tripartite set of conditions, and shows how this applies to cases of the evolution of complexity at higher-levels of the biological hierarchy. CNE was initially deployed to help explain a variety of complex molecular structures and processes, including spliceosomal splicing, trypansomal pan-editing, scrambled genes in ciliates, duplicate gene retention and fungal (...) ATP synthetase structure. CNE has also been generalized to apply to phenomena at the cellular level, including protein-protein interaction network modularity, obligate microbial symbioses, eukaryogenesis and microbial unculturability. This essay further extends CNE to cases of complexity at levels of organization higher than the molecular and cellular. These are multicellular phenotypic complexity, multicellular ecological complexity and, some cases of cultural complexity. (shrink)