Despite its many extensions and implications, we argue that punctuated equilibrium itself has two core, empirical claims: (1) stasis dominates within fossil species; and (2) morphological change is concentrated in pulses that occur associated with speciation. Here we assess the state of the evidence for these two claims, 50 years after punctuated equilibrium’s foundational paper. Spurred by controversy, paleontologists have amassed a large number of case studies in which morphology in species-level lineages is tracked over time. Modern, likelihood-based methods have been used to fit to these data models of stasis, random walks, and directional trends, as well as more complex dynamics. Compilations reveal that the directional trends predicted by gradualist expectations are infrequent. Although stasis is commonly observed, it is favored in less than half of cases, and meandering random walks or more complex models generally account for the majority of cases. The second claim of punctuated equilibrium has received much less empirical scrutiny than the first. Although speciational pulses are plausible in theory, only a few paleontological studies integrate ancestor–descendant time series into a phylogenetic framework as is needed to estimate cladogenetic change and compare it with anagenesis. These studies, as well as more indirect analyses of extant clades, suggest that speciational change can occur, but we cannot yet assess with confidence its frequency or importance compared with anagenetic changes.

The nature of phenotypic evolution within lineages is central to many unresolved questions in paleontology and evolutionary biology. Analyses of evolutionary time series of ancestor–descendant populations in the fossil record are likely to make important contributions to many of these debates. However, the limited number of models that have been applied to these types of data may restrict our ability to interpret phenotypic evolution in the fossil record. Using uni- and multivariate models of trait evolution that make different assumptions regarding the dynamics of the adaptive landscape, I evaluate contrasting hypotheses to explain evolution of size in the radiolarian Eucyrtidium calvertense and armor in the stickleback Gasterosteus doryssus. Body-size evolution in E. calvertense is best explained by a model in which the lineage evolves as a consequence of a shift in the adaptive landscape that coincides with the initiation of neosympatry with its sister lineage. Multivariate evolution of armor traits in a stickleback lineage (G. doryssus) shows evidence of adaptation toward independent optima on the adaptive landscape at the same time as traits change in a correlated fashion. The fitted models are available in the R package evoTS, which builds on the paleoTS framework.