Abstract The three extant Divisions comprising the bryophytes extend, as fossils, well back into Palaeozoic time. Bryophyte origin is part of the rise of terrestrial, vascularized, plants with sporopollenin-walled spores in the Silurian. Before the end of Carboniferous time, bryophyte lines were widely present. Separation of Gondwana and Laurasia by the Permian Tethys Sea and subsequent widespread desert episodes fragmented an already diversified bryoflora subjecting it to intense selective pressure. The cool, mesic climate of southern Gondwana provided a refugium for austral bryophytes. Warmer and drier climates of the Permo-Triassic Laurentian-Laurasia favoured drought-adapted or niche-specific groups creating marked systematic discontinuities. The Angaran wet, probably cool, temperate region provided refuge for basic stock for much of today's rich holarctic and wet ‘tropical’ bryofloras. Climatic changes, correlated with tectonic events and the rise of angiosperms, opened habitats favourable for a diversity explosion. Despite demonstrated potential for long-distance dispersal, modern distributions are mostly linked with total floras or establishment on islands prior to niche saturation. Remnants of Gondwanan bryoflora persist in high southern latitudes as disjunctions with the possibility that the folded ranges of the African Cape have been an insular fragment at higher latitudes becoming attached shortly after angiosperm diversification. Floras of southern India and east Africa have common features but the Himalayan flora shows evidence that the Gondwanan flora of the Indian plate was lost during the movement through desert and tropical latitudes; neotropical and palaeotropical floras are distinctive. Much of the northern Australian bryoflora is recently Malesian-derived while the southeast shows strong austral influence and commonality with New Zealand. Tropical Pacific island floras are mostly Malesian-derived but with both holarctic and austral elements present as in Hawaii and the Society Islands. Holarctic bryoflora is circum-polar with temperate areas of Euro-American and far eastern elements floristically bound by disjunct and vicariad species. Kroeber Coefficients of Correlation differ as Pacific island floras are compared and Guttman-Lingoes Smallest Space Coordinates indicates floristic subgroups within Polynesia. Although these and other mathematical treatments yield potentially promising results, the methods are yet unrefined and there is some uncertainty whether characteristics of numbers or of organisms are implicit in the summations.