The frequent usage of figurative language on online social networks, especially on Twitter, has the potential to mislead traditional sentiment analysis and recommender systems. Due to the extensive use of slangs, bashes, flames, and non-literal texts, tweets are a great source of figurative language, such as sarcasm, irony, metaphor, simile, hyperbole, humor, and satire. Starting with a brief introduction of figurative language and its various categories, this article presents an in-depth survey of the state-of-the-art techniques for computational detection of seven different figurative language categories, mainly on Twitter. For each figurative language category, we present details about the characterizing features, datasets, and state-of-the-art computational detection approaches. Finally, we discuss open challenges and future directions of research for each figurative language category.