The number of mobile users grows constantly; an important demand is for IT to be accessible always and everywhere. Users have gone from being stationary to mobile, accessing and using their electronic services everywhere, even while moving. But not only users can be mobile; when migrating an application from one device to another, one has achieved application mobility. As migration is often carried out in heterogeneous environments, applications have to be able to receive and manage input from the user and the surroundings. Taking interfaces and gathered information about the context (defined as any information that can be used to characterize the situation of an entity) with them while migrating, the notion of context-awareness supported application mobility has become a reality, and this would indeed enhance the field of ubiquitous computing, where the vision is often hidden, yet always accessible applications. The purpose of this survey is twofold: to compile previous work on context-awareness supported application mobility and to identify challenges connected to the area. The challenges that are identified concern how to identify and keep track of applications that move between devices; how to best design the model for context-awareness; how to attain and communicate a good enough context quality; how to achieve seamlessness and minimize downtime while executing a migration; and finally how to handle heterogeneity within and between devices, in shape of differences in hardware and software.