With the proliferation of the Internet of Things (IoT) and the development of wireless communication technologies such as 5G, new types of services are emerging and mobile data traffic is growing exponentially. The mobile computing model has shifted from traditional cloud computing to mobile edge computing (MEC) to ensure QoS. The main feature of MEC is to “sink” network resources to the edge of the network to meet the needs of delay-sensitive and computation-intensive services, and to provide users with better services. Computation offloading is one of the major research issues in MEC. In this paper, we summarize the state of the art in task offloading in MEC. First, we introduce the basic concepts and typical application scenarios of MEC, and then we formulate the task offloading problem. In this paper, we analyze and summarize the state of research in the industry in terms of key technologies, schemes, scenarios, and objectives. Finally, we provide an outlook on the challenges and future research directions of computational offloading techniques and indicate the suggested direction of follow-up research work.