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Abstract Phylogenetic comparative methods are a major tool for evaluating macroevolutionary hypotheses. Methods based on the mean-reverting stochastic Ornstein–Uhlenbeck process allow for modelling adaptation on a phenotypic adaptive landscape that itself evolves and where fitness peaks depend on measured characteristics of the external environment and/or other organismal traits. Here, we give an overview of the conceptual framework for the many implementations of these methods and discuss how we might interpret estimated parameters. We emphasize that the ability to model a changing adaptive landscape sets these methods apart from other approaches and discuss why this aspect captures long-term trait evolution more realistically. Recent multivariate extensions of these methods provide a powerful framework for testing evolutionary hypotheses but are also more complicated to use and interpret. We provide some guidance on their usage and put recent literature on the topic in biological rather than mathematical terms. We further show how these methods provide a starting point for modelling reciprocal selection (i.e., coevolution) between interacting lineages. We then briefly review some critiques of the methodologies. Finally, we provide some ideas for future developments that we think will be useful to evolutionary biologists. 

Abstract Models based on the Ornstein–Uhlenbeck process have become standard for the comparative study of adaptation. Cooper et al. (2016) have cast doubt on this practice by claiming statistical problems with fitting Ornstein–Uhlenbeck models to comparative data. Specifically, they claim that statistical tests of Brownian motion may have too high Type I error rates and that such error rates are exacerbated by measurement error. In this note, we argue that these results have little relevance to the estimation of adaptation with Ornstein–Uhlenbeck models for three reasons. First, we point out that Cooper et al. (2016) did not consider the detection of distinct optima (e.g. for different environments), and therefore did not evaluate the standard test for adaptation. Second, we show that consideration of parameter estimates, and not just statistical significance, will usually lead to correct inferences about evolutionary dynamics. Third, we show that bias due to measurement error can be corrected for by standard methods. We conclude that Cooper et al. (2016) have not identified any statistical problems specific to Ornstein–Uhlenbeck models, and that their cautions against their use in comparative analyses are unfounded and misleading. [adaptation, Ornstein–Uhlenbeck model, phylogenetic comparative method.]