While empathy is a century-old psychological concept, its study in non-human animals has become the focus of much recent scientific interest, as it promises to provide the clues to understand the evolutionary origins of our social and moral nature. A review of the comparative study of empathy is thus timely to complement and constrain anthropocentric views, and to integrate current findings. However, this is not an easy task. The study of animal empathy has developed using different paradigms, different concepts of the phenomena involved, and the absence of a systematic program. Herein, we carry out a comprehensive review of the literature on complex forms of empathy in non-human animals: sympathetic concern and empathic perspective-taking. In particular, we focus on consolation and targeted helping, as the best examples of each category. In so doing, we try to shed light on the current debate concerning whether these phenomena are exclusively human traits. First, we try to clarify the terminology and taxonomy of forms of empathy, providing operative criteria for these phenomena that are applicable to both human and non-human animals. Second, we discuss whether the available evidence qualifies such behaviour as empathic. Third, we aim to provide an integrative view of the field, clarifying the challenges and conditions to satisfy. We also hope to highlight the importance of the study of these processes for elucidating the evolutionary history of this capacity across the animal kingdom.