Most traits expressed by organisms, such as life-history traits, gene expression profiles, developmental trajectories, behavioural sequences and reaction norms are function-valued traits (colloquially “phenotypically plastic traits”), since they vary across an individual’s age and in response to various internal and/or external factors (state variables). Furthermore, most organisms live in populations subject to limited genetic mixing and are thus likely to interact with their relatives. I show how to characterise selection on genetically determined function-valued traits of individuals interacting in a group-structured population, by deriving the marginal version of Hamilton’s rule for function-valued traits. This rule simultaneously gives a condition for the invasion of an initially rare mutant function-valued trait and its ultimate fixation in the population (invasion thus implies substitution). Hamilton’s rule thus underlies the gradual evolution of function-valued traits and gives rise to necessary first-order conditions for their uninvadability (evolutionary stability). I will briefly discuss recent developments on second-order conditions and conditions for convergence stability. The analysis combines optimal control theory and differential game theory, to simultaneously characterize selection on (i) open-loop traits – functions of time (or age) only, and (ii) closed-loop (state-feedback) traits – functions of both time and state variables. The aim of the work is to delineate the role of state-dependence and interdependence between individuals for trait evolution, which has implications to both life-history theory and social evolution.