This chapter develops a theory of Homeric creativity that is informed by both contemporary cognitive studies and by ancient ideas about poetic craft and divine inspiration, as seen through the lenses of archaic poetry and Plato’s Ion. Sections 1-2 survey the conceptions of the poet as a craftsman vs. divinely inspired in Homer, Hesiod, Archilochus, Pindar, Democritus, and Plato. Section 3 introduces the modern study of creativity in the field of cognitive psychology, the study of jazz improvisation, and the concept of flow. Section 4 introduces the neuroscience of improvisation, including recent fMRI studies on jazz and lyric improvisation. The results of these studies (pointing to a state of hypofrontality when carrying out complex improvisational tasks) are compatible with some ancient ideas on creativity and inspiration (such as the idea that an external source - like the Muses- might be involved in these creative acts).