Comparative studies of large phylogenies of living and extinct groups have shown that most biodiversity arises from a small number of highly species-rich clades. To understand biodiversity, it is important to examine the history of these clades on geological time scales. This is part of a distinct 'phylogenetic expansion' view of macroevolution, and contrasts with the alternative, non-phylogenetic 'equilibrium' approach to the history of biodiversity. The latter viewpoint focuses on density-dependent models in which all life is described by a single global-scale model, and a case is made here that this approach may be less successful at representing the shape of the evolution of life than the phylogenetic expansion approach. The terrestrial fossil record is patchy, but is adequate for coarse-scale studies of groups such as vertebrates that possess fossilizable hard parts. New methods in phylogenetic analysis, morphometrics and the study of exceptional biotas allow new approaches. Models for diversity regulation through time range from the entirely biotic to the entirely physical, with many intermediates. Tetrapod diversity has risen as a result of the expansion of ecospace, rather than niche subdivision or regional-scale endemicity resulting from continental break-up. Tetrapod communities on land have been remarkably stable and have changed only when there was a revolution in floras (such as the demise of the Carboniferous coal forests, or the Cretaceous radiation of angiosperms) or following particularly severe mass extinction events, such as that at the end of the Permian.