Recent findings in cognitive neuroscience shed light on the existence of a common neural mechanism that could account for action and intention to understand abilities in humans and non-human primates. Empirical evidence on the neural underpinnings of action goals and on their ontogeny and phylogeny is introduced and discussed. It is proposed that the properties of the mirror neuron system and the functional mechanism describing them, embodied simulation, enabled pre-linguistic forms of action and intention understanding. Basic aspects of social cognition appear to be primarily based on the motor cognition that underpins one's own capacity to act, here defined as motor abstraction. On the basis of this new account of the motor system, it is proposed that intersubjectivity is the best conceived of as intercorporeity.