The directly attached diploporite cystoid Sphaeronites is characterized by a closely approximated mouth and anus on an oral platform, and densely developed diplopores. Three subgenera are recognized: Sphaeronites s.s. (lower Ordovician, Sweden); Peritaphros (middle and upper Ordovician, Britain, Norway, Sweden, Baltic States); Ataphros subgen. nov. (upper Ordovician, South Wales). Early species occur in 'Sphaeronites beds' with few other echinoderms. Later species are less abundant, but occur with a more diverse echinoderm fauna. Sphaeronites shows evolutionary trends towards increased plate thickness and plate size (hence reduced number of plates per theca), increased anal area (and inferred food processing capacity), and decreased diplopore density. Diplopore structure also evolved. Those of Sphaeronites s.s. are deeply sunken with blunt spines on the rims; Peritaphros diplopores have a large central tubercle surrounded by a moat-like channel but no rim spines; Ataphros diplopores have just the central tubercle. Changes in diplopores have been investigated using computer programs to simulate respiratory gas exchange. The larger, less densely packed diplopores of later species achieved the same respiratory capacity, but with individually more efficient diplopores. Reconstructions of the soft tissue cover of diplopores confined to the moat-like channels of Peritaphros or covering the central tubercle as well, yielded respiratory capacities 10% and 100% of Sphaeronites s.s. The maximum calculated evolutionary increase in saturation with O2 of body fluids occurred with soft tissue 'covers 20 ^m thick. It is inferred that these covers were 10-50 /*m thick in life. Calculation of food processing capacity suggests that Sphaeronites could gather all the food they could metabolize with simple, unbranched appendages (?brachioles) of a length equal to the thecal diameter. Computer programs offer a powerful tool in reconstructing unpreserved parts of fossils essential to functional interpretations. Sphaeronites is revised systematically. Nine species are recognized of which one is left under open nomenclature and the following are new: S. (Sphaeronites) minor, S. (Peritaphros) pauciscleritatus, S. (P.) variabilis, and S. (Ataphros) insuliporus. Sphaeronites was a directly attached diploporite cystoid genus which survived throughout the Ordovician from the Arenig to the Ashgill and is found in Britain, Scandinavia, and the Baltic States. It is type genus of the family Sphaeronitidae and superfamily Sphaeronitida. The type species of Sphaeronites, Echinus pomum Gyllenhaal, 1772, was one of the first two cystoids ever described. Historically and taxonomically Sphaeronites is an important genus. In this paper we revise the genus, document its evolutionary trends, and show that Sphaeronites is also very important in an evolutionary context.We first became interested in the genus independently, in the course of revising British (CRCP) and Norwegian (JFB) Ordovician cystoid faunas, respectively. Also independently, we realized that the Swedish cystoid figured by Regnell (1945, pi. 15, fig. 1) as 'Gen. & sp. indet.' was a new species of Sphaeronites. On discovering our joint interest we combined to revise the genus thoroughly in Britain and Scandinavia, considering its stratigraphical and geographical occurrences, functional morphology, mode of life, and evolution. Paul (1972, 1973) had already documented an evolutionary trend in diplopore density, and the functional significance of this trend has been investigated quantitatively using computer programs to simulate gas exchange. As a result of this study we recognize eight named species, four new, in three subgenera, one new. The evolutionary trend in diplopore density achieved the same respiration rate with progressively fewer, but individually more efficient, diplopores. Computer-generated quantitative estimates of respiration rates have also enabled us to reconstruct some unpreserved parts of the fossils, namely the cover of the diplopores and feeding.