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Mobile devices supporting the "Internet of Things" (IoT), often have limited capabilities in computation, battery energy, and storage space, especially to support resource-intensive applications involving virtual reality (VR), augmented reality (AR), multimedia delivery and artificial intelligence (AI), which could require broad bandwidth, low response latency and large computational power. Edge cloud or edge computing is an emerging topic and technology that can tackle the deficiency of the currently centralized-only cloud computing model and move the computation and storage resource closer to the devices in support of the above-mentioned applications. To make this happen, efficient coordination mechanisms and “offloading” algorithms are needed to allow the mobile devices and the edge cloud to work together smoothly. In this survey paper, we investigate the key issues, methods, and various state-of-the-art efforts related to the offloading problem. We adopt a new characterizing model to study the whole process of offloading from mobile devices to the edge cloud. Through comprehensive discussions, we aim to draw an overall “big picture” on the existing efforts and research directions. Our study also indicates that the offloading algorithms in edge cloud have demonstrated profound potentials for future technology and application development.