The identification of any form of social learning, imitation, copying or mimicry presupposes a notion of correspondence between two autonomous agents. Judging whether a behavior has been transmitted socially requires the observer to identify a mapping between the demonstrator and the imitator. If the demonstrator and imitator have similar bodies, e.g. are animals of the same species, of similar age, and of the same gender, then to a human observer an obvious correspondence is to map the corresponding body parts: left arm of demonstrator maps to left arm of imitator, right eye of demonstrator maps to right eye of imitator, tail of demonstrator maps to tail of imitator. There is also an obvious correspondence of actions: raising the left arm by the model corresponds to raising the left arm by the imitator, production of vocal signals by the model corresponds to the production of acoustically similar ones by the imitator, picking up a fruit by the demonstrator corresponds to picking up a fruit of the same type by the imitator. Furthermore, there is a correspondence in sensory experience: audible sounds, a touch, visible objects and colors, and so on evidently seem to be detected and experienced in similar ways. What to take as the correspondence seems relatively clear in this case. As humans, we are good at imitating and at recognizing such correspondences. It is also clear that most other animals, robots, and software programs may in fact generally fail to recognize any such correspondences. To judge a produced behavior to be a copy of an observed one, we require at least that it respects some such correspondence. The faithfulness or precision of the behavioral match can obviously vary, and no absolute cutoff or threshold exists defining success as opposed to failure of behavioral matching. But one can study the degree of success using various metrics and measures of correspondence (Nehaniv & Dautenhahn, 2001; also see below). Moreover, it turns out that the obvious correspondences between similar bodies mentioned above are not the only ones possible. Consider a human imitating another one that is facing her: if the demonstrator raises her left arm, should the imitator raise her own left arm? Or should she raise her right, to make a "mirror image" of the demonstrator's actions? If the demonstrator picks up a brush, should an imitator pick up the same brush? Or just another brush of the same type? If the demonstrator opens a container to get at chocolate inside, should the imitator open a similar container in the same way e.g. by unwrapping but not tearing the surrounding paper?, or is it enough just to open the container somehow? The different possible answers to these questions presuppose different correspondences. If a child watches a teacher solving subtraction problems in arithmetic, and then solves for the first time similar but not identical problems on its own, social learning has occurred. But what type of correspondence is at work here? In China and Japan, the ideographic character for to imitate also means to learn or to study. By going through the motions of an algorithm for solving sample problems, students everywhere are able to learn how to solve similar ones, of course without necessarily gaining understanding of why the procedures they have learned work. In this chapter, for lack of a better term, we shall use the word imitator to refer to any autonomous agent performing a candidate behavioral match. The use of this word here does not entail any particular mechanism of matching or any particular type of social learning. In what follows, we shall describe how different matching phenomena arise depending on the criteria employed in generating the behavior of the imitator. For example, goal emulation, stimulus enhancement, mimicry, and so on, will all be cast as solutions to correspondence problems with different particular selection criteria.