Abstract This chapter proposes to define metaphor as a visual-material structure, the sphere of which is ontological rather than cognitive or conceptual. It argues that the essence of metaphor, as either an aesthetic or a communicative unit or both, resides in the qualitative dimension and appearance, or even materiality, of the metaphorical medium and its form. The chapter thus offers a new theory of metaphor, focusing on the medium of metaphor, which composes and transfigures or reconstructs its target anew: a composition that is prior to understanding or conceptualizing the target. In doing so, the chapter presents a formalist ontology of metaphors, established via an externalist account of metaphors, as opposed to the prevailing cognitivistconceptualist account which I characterize as internalist. The various kinds of metaphor – linguistic, poetic, visual or material – are based on their external structure, rather than an internal-conceptual mechanism of understanding, as assumed by a significant segment of the literature. Visual metaphor, therefore, is the paradigmatic kind of metaphor, the analysis of which can be generalized to other kinds of metaphor. Furthermore, the chapter tries to overcome the current discrepancy between the formalist character of the metaphorical medium and the dominance of cognitivist and conceptualist theories of metaphor. Challenging these, I claim that if the identity of metaphor is indeed based on its composition, then it is actually based on its aesthetic qualities. That is to say, not only are there autonomous visual or material metaphors, that are not based on linguistic or conceptual ones, but linguistic and conceptual metaphors are based on visuality: they are enabled by the structural possibilities offered by visual media.