Abstract: The long-term diversification of life probably cannot be modelled as a simple equilibrial process: the time scales are too long, the potential for exploring new ecospace is too large and it is unlikely that ecological controls can act at global scales. The sum of many clade expansions and reductions, each of which happens according to its own dynamic, probably approximates more a damped exponential curve when translated into a global-scale species diversification curve. Unfortunately, it is not possible to plot such a meaningful global-scale species diversification curve through time, but curves at higher taxonomic levels have been produced. These curves are subject to the vagaries of the fossil record, but it is unlikely that the sources of error entirely overwhelm the biological signal. Clades radiate when the external and internal conditions are right: a new territory or ecospace becomes available, and the lineage has acquired a number of characters that open up a new diet or mode of life. Modern high levels of diversity in certain speciose clades may depend on such ancient opportunities taken. Dramatic climatic changes through the Quaternary must have driven extinctions and originations, but many species responded simply by moving to more favourable locations. Ecological communities appear to be no more than merely chance associations of species, but there may be real interactions among species. Ironically, high species diversity may lead to more speciation, not, as had been assumed, less: more species create more opportunities and selective pressures for other species to respond to, rather than capping diversity at a fixed equilibrium level. Studies from the scale of modern ecosystems to global long-term patterns in the fossil record support a model for the exponential diversification of life, and one explanation for a pattern of exponential diversification is that as diversity increases, new forms become ever more refinements of existing forms. In a sense the world becomes increasingly divided into finer niche space. Organisms have a propensity to speciate freely, species richness within ecosystems appears to generate opportunities for more speciation, clades show all kinds of patterns from sluggish speciation rates and constant diversity through time to apparently explosive speciation, and there is no evidence that rapidly speciating clades have reached a limit, nor that they are driving other clades to extinction. A corollary of this view is that current biodiversity must be higher than it has ever been. Limits to infinite growth are clearly local, regional, and global turnover and extinction events, when climate change and physical catastrophes knock out species and whole clades, and push the rising exponential curve down a notch or two.