This paper provides a comprehensive review of the emerging research area of mobile molecular communication. In mobile molecular communication, sender and receiver bionanomachines as well as associated nodes in the environment exhibit dynamic behavior in the sense that they are mobile and communicate while they move. This paper presents a model of mobile bionanomachines and uses the model to discuss how groups of such bionanomachines working in unison can provide useful functionalities. This paper illustrates several functionalities by applying mobile molecular communication to the concept of cooperative drug delivery. Unsolved research challenges in this area are outlined and discussed.