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Edge-cloud computing, combining the benefits of both edge computing and cloud computing, is one of the most promising ways to address the resource insufficiency of smart devices. Task offloading is an important challenge must be addressed for edge-cloud computing in practice, which decides the place and the time for performing each task. Even though there is existing research focusing on the task offloading in edge-cloud computing, a lot of problems should be solved before the application of these offloading technologies. Thus, in this article, we first propose a taxonomy of task offloading in edge-cloud environments to investigate and classify related research articles, and then summarize several challenges which have not been addressed for future research directions on this area to promote the development of edge-cloud market.