Previous work on the coverage of mobile sensor networks focuses on algorithms to reposition sensors in order to achieve a static configuration with an enlarged covered area. In this paper, we study the dynamic aspects of the coverage of a mobile sensor network that depend on the process of sensor movement. As time goes by, a position is more likely to be covered; targets that might never be detected in a static sensor network can now be detected by moving sensors. We characterize the area coverage at specific time instants and during time intervals, as well as the time it takes to detect a randomly located static target. Our results show that sensor mobility can be exploited to compensate for the lack of sensors and improve the network coverage. For mobile targets, we take a game theoretic approach and derive optimal mobility strategies for sensors and the targets from their own perspectives.