Metaphor is an indispensable part of human cognition and everyday communication. Much research has been conducted elucidating metaphor processing in the mind/brain and the role it plays in communication. In recent years, metaphor processing systems have benefited greatly from these studies, as well as the rapid advances in deep learning for natural language processing (NLP). This paper provides a comprehensive review and discussion of recent developments in automated metaphor processing, in light of the findings about metaphor in the mind, language, and communication, and from the perspective of downstream NLP tasks.