Metaphors are figurative expressions frequently appearing daily. Given its significance in downstream natural language processing tasks such as machine translation and sentiment analysis, computational metaphor processing has led to an upsurge in the community. The progress of Artificial Intelligence has incentivized several technological tools and frameworks in this domain. This article aims to comprehensively summarize and categorize previous computational metaphor processing approaches regarding metaphor identification, interpretation, generation, and application. Though studies on metaphor identification have made significant progress, metaphor understanding, conceptual metaphor processing, and metaphor generation still need in-depth analysis. We hope to identify future directions for prospective researchers based on comparing the strengths and weaknesses of the previous works.