Following a lead from botanists, this paper offers a re-interpretation of the morphology and growth of scleractinian, rugosan and tabulate corals in terms of iterated morphological units (modules), with an emphasis on the shape and organization of colonies. It presents a new theoretical basis for modules, a new descriptive framework for corals above the zooidal level, a review of practical applications and a comparison of coral modules with other kinds of organic iteration. Consideration of cloning is a prerequisite and leads to a revival of the hypothesis that colonies have arisen by arrested budding (clonoteny), where zooids are paedomorphic to varying degrees with respect to the primitive state of clonally produced, detached (clonoparous) individuals. A second hypothesis follows, that the mouth structure is universally homologous in all corals. R. Riedl's work (Order in living organisms. Chichester: John Wiley (1978)) on morphological organization and hierarchy can therefore be applied to corals to obtain a non-arbitrary theoretical basis for recognizing modules and distinguishing them from other kinds of repetition of parts. There are five criteria for modularity, four topological and one homological: (i) a modular structure is a three-dimensional tesselation; (ii) modules correspond to homogeneous units of hierarchical subdivision; (iii) an organism is modular if subdivision at the highest (or very high) level reveals homogeneous units; (iv) if homogeneous units occur in an unbroken series of subdivisions, they are all modules at their respective levels (hence modules of modules, that is, cormidia), and the finest unit is the fundamental module; (v) homogeneous units are those which are homologous with each other (homonoms). On this basis, and allowing for interzooidal connective structures, the fundamental module of corals is the zooid (not the polyp alone). Differences between zooids, as suggested by other authors, are of degree, not kind. Colony form can therefore be specified by using a scheme that starts with zooids, and is based on (i) organization (component morphology and modular arrangement), and (ii) shape, for every modular level present. The basis for higher level modules is reiterative, either homomodular, or heteromodular (subdivided into polymorphic or polystatic). Branching organization and branching shape should be distinguished, and either can occur at any modular level, or even not at all. Modules bring potentially greater precision to growth studies, statistically and topologically, either heuristically or to test hypotheses about fecundity, senescence, determinate growth, variation and inherited architecture. Higher level modularity has previously been largely ignored. Density banding, dye markers and computer modelling, ideally in combination, are likely to produce the most useful results in the near future, and should lead to advances in taxonomy and phylogenetics, in the understanding of the morphological consequences of the coral-zooxanthellaesymbiosis, and in the inference of past conditions from fossil corals. In a wider phyletic context, zooidal (= fundamental) modules are meristematic and broadly metameric, though it follows from R. B. Clark (Zool. Jb. 103, 169-195 (1980)) that this does not automatically signify a close phylogenetic relationship with annelids, or with any other metameric organisms. Coral 'metamers' are probably non-serial and clonal because the coral mode of life is usually vegetative. They exist at a higher organizational level than these other metamers.