Biological classification has been successively shaped by Neoplatonism, ideas of plenitude and one-dimensional continuity (the Great Chain of Being), two-dimensional continuity (Linnaeus's map), idealized comparative morphology and development and, through phylogenetic theory, Darwinian descent with modification. Concepts of Algae have thus evolved within a succession of very different paradigms. Algae have been imperfect plants; a segment of the Great Chain; the least-developed members of vegetative form-series; and primitive forms from which plants (and even animals) can be elaborated. The idea of a single grouping of simple organisms, formalized by Bory de Saint-Vincent and others in the early nineteenth century, was developed in a phylogenetic framework by Haeckel as Protista. Most macrophytic algae, however, remained Plants until Copeland merged eukaryotic non-green algae into Protista. Arrangements of algal taxa within form-series, and recognition of Algae and Protista as taxa, are incompatible with Hennigian phylogenetic systematics (cladistics). Developing genomic perspectives threaten to undermine established concepts of organismal lineage and higher-level biological classification.