Irony is a ubiquitous figurative language in daily communication. Previously, many researchers have approached irony from linguistic, cognitive science, and computational aspects. Recently, some progress have been witnessed in automatic irony processing due to the rapid development in deep neural models in natural language processing (NLP). In this paper, we will provide a comprehensive overview of computational irony, insights from linguisic theory and cognitive science, as well as its interactions with downstream NLP tasks and newly proposed multi-X irony processing perspectives.