In this paper, we review reports and present new empirical data from studies with marmosets and dogs that address the correspondence problem of imitation research. We focus on the question of how it is possible to transform visual information into matching motor acts. Here, the important issue is not the learning of a complex skill, but determining the copying fidelity of animals at different levels of behavioural organization. As a theoretical framework, we suggest a classification in terms of movement, action and result, which shows a positive relationship between the organizational level of imitation and matching degree. While the monkey studies have provided evidence of very precise copying of movements and, to a lesser degree, of behaviours, the dog studies have provided evidence of action copying and the reproduction of results. In a Do-as-I-do study, a dog attempted to reproduce the results of demonstrated object manipulations at the expense of movement details. Transitive actions were more easily replicated than intransitive ones, and familiarity of actions had a major influence. The discussion of these findings addresses the question of the neuronal mechanisms underlying imitation and whether a single mechanism is sufficient to explain the different levels of copying fidelity.