Many central questions in ecology and evolutionary biology require characterizing phenotypes that change with time and environmental conditions. Such traits are inherently functions, and new 'function-valued' methods use the order, spacing, and functional nature of the data typically ignored by traditional univariate and multivariate analyses. These rapidly developing methods account for the continuous change in traits of interest in response to other variables, and are superior to traditional summary-based analyses for growth trajectories, morphological shapes, and environmentally sensitive phenotypes. Here, we explain how function-valued methods make flexible use of data and lead to new biological insights. These approaches frequently offer enhanced statistical power, a natural basis of interpretation, and are applicable to many existing data sets. We also illustrate applications of function-valued methods to address ecological, evolutionary, and behavioral hypotheses, and highlight future directions.