The question of how much the outcomes of cultural evolution are shaped by the cognitive capacities of human learners has been explored in several disciplines, including psychology, anthropology and linguistics. We address this question through a detailed investigation of transmission chains, in which each person passes information to another along a chain. We review mathematical and empirical evidence that shows that under general conditions, and across experimental paradigms, the information passed along transmission chains will be affected by the inductive biases of the people involved-the constraints on learning and memory, which influence conclusions from limited data. The mathematical analysis considers the case where each person is a rational Bayesian agent. The empirical work consists of behavioural experiments in which human participants are shown to operate in the manner predicted by the Bayesian framework. Specifically, in situations in which each person's response is used to determine the data seen by the next person, people converge on concepts consistent with their inductive biases irrespective of the information seen by the first member of the chain. We then relate the Bayesian analysis of transmission chains to models of biological evolution, clarifying how chains of individuals correspond to population-level models and how selective forces can be incorporated into our models. Taken together, these results indicate how laboratory studies of transmission chains can provide information about the dynamics of cultural evolution and illustrate that inductive biases can have a significant impact on these dynamics.