Sensory and emotional experiences such as pain and empathy are essential for mental and physical health. Cognitive neuroscience has been working on revealing mechanisms underlying pain and empathy. Furthermore, as trending research areas, computational pain recognition and empathic artificial intelligence (AI) show progress and promise for healthcare or human-computer interaction. Although AI research has recently made it increasingly possible to create artificial systems with affective processing, most cognitive neuroscience and AI research do not jointly address the issues of empathy in AI and cognitive neuroscience. The main aim of this paper is to introduce key advances, cognitive challenges and technical barriers in computational pain recognition and the implementation of artificial empathy. Our discussion covers the following topics: How can AI recognize pain from unimodal and multimodal information? Is it crucial for AI to be empathic? What are the benefits and challenges of empathic AI? Despite some consensus on the importance of AI, including empathic recognition and responses, we also highlight future challenges for artificial empathy and possible paths from interdisciplinary perspectives. Furthermore, we discuss challenges for responsible evaluation of cognitive methods and computational techniques and show approaches to future work to contribute to affective assistants capable of empathy.