Interesting systems, whether biological or artificial, develop. Starting from some initial conditions, they respond to environmental changes, and continuously improve their capabilities. Developmental psychologists have dedicated significant effort to studying the developmental progression of infant imitation skills, because imitation underlies the infant's ability to understand and learn from his or her social environment. In a converging intellectual endeavour, roboticists have been equipping robots with the ability to observe and imitate human actions because such abilities can lead to rapid teaching of robots to perform tasks. We provide here a comparative analysis between studies of infants imitating and learning from human demonstrators, and computational experiments aimed at equipping a robot with such abilities. We will compare the research across the following two dimensions: (a) initial conditions--what is innate in infants, and what functionality is initially given to robots, and (b) developmental mechanisms--how does the performance of infants improve over time, and what mechanisms are given to robots to achieve equivalent behaviour. Both developmental science and robotics are critically concerned with: (a) how their systems can and do go "beyond the stimulus" given during the demonstration, and (b) how the internal models used in this process are acquired during the lifetime of the system. (Contains 2 figures.)