Empathy is an important psychological process that facilitates human communication and interaction. Enhancement of empathy has profound significance in a range of applications. In this paper, we review emerging directions of research on computational analysis of empathy expression and perception as well as empathic interactions, including their simulation. We summarize the work on empathic expression analysis by the targeted signal modalities (e.g., text, audio, facial expressions). We categorize empathy simulation studies into theory-based emotion space modeling or application-driven user and context modeling. We summarize challenges in computational study of empathy including conceptual framing and understanding of empathy, data availability, appropriate use and validation of machine learning techniques, and behavior signal processing. Finally, we propose a unified view of empathy computation, and offer a series of open problems for future research.