A relationship between the rate of molecular change and diversification has long been discussed, on both theoretical and empirical grounds. However, the effect on our understanding of evolutionary patterns is yet to be fully explored. Here we develop a new model, the Covariant Evolutionary Tempo (CET) model, with the aim of integrating patterns of diversification and molecular evolution within a framework of a continuously changing ‘tempo’ variable that acts as a master control for molecular, morphological and diversification rates. Importantly, tempo itself is treated as being variable at a rate proportional to its own value. This model predicts that diversity is dominated by a small number of extremely large clades at any historical epoch including the present; that these large clades are expected to be characterised by explosive early radiations accompanied by elevated rates of molecular evolution; and that extant organisms are likely to have evolved from species with unusually fast evolutionary rates. Under such a model, the amount of molecular change along a particular lineage is essentially independent of its height, which weakens the molecular clock hypothesis. Finally, our model explains the existence of ‘living fossil’ sister groups to large clades that are species poor and exhibit slow rates of morphological and molecular change. Our results demonstrate that the observed historical patterns of evolution can be modelled without invoking special evolutionary mechanisms or innovations that are unique to specific times or taxa, even when they are highly non-uniform.