The cognitive linguistic approach to metaphor and metonymy postulates that we think and understand the world with the help of these figurative mechanisms. Even though metaphors are thus not primarily figures of speech, their visual representatives have not been given enough attention in terms of academic research. The study of conceptual metonymies is even more bridled with the lack of scholarly attention. In recent years, however, linguists have focused on nonverbal metaphors, analyzing visual arts, advertising, and film. Numerous genres in art have shown traits of metaphoric expression to a greater or lesser extent, the most noticeable among them being surrealism as an artistic movement with a highly potent metaphoric system. In an effort to discover the conceptual metaphors and metonymies in the still and moving images (visual art and film), this dissertation will attempt to shed a light on visual metaphor and metonymy in Surrealist art - more specifically, the art of H.R. Giger, Swiss Surrealist whose creations, examined through the lens of semiotics as a highly complex, multimodal structure, surpass cultural and sociological barriers and point to the merger between man and technology, indicative of the modern transformational processes of the 20th and the 21st century.