Mobile opportunistic networks (MONs) utilize the communication opportunities arising from node contacts to forward data in a hop-by-hop manner, and they play an important role in implementing the intensive perception and ubiquitous interconnection in Internet of Things. Opportunistic routing is the basic method for data communication in intermittently connected scenarios, so it is worth studying and has captured great interests from researchers. First, this paper introduces the concept, system architecture, typical applications of MONs, and some technical challenges in this field. Second, it reviews the opportunistic routing problem from three aspects: 1) evaluation metrics, 2) design requirements, 3) forwarding schemes. Some of the latest progresses of this research are also presented. Finally, the future research trends of opportunistic routing are analyzed and prospected.