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With the continuous growth of mobile data and the unprecedented demand for computing power, resource-constrained edge devices cannot effectively meet the requirements of Internet of Things (IoT) applications and Deep Neural Network (DNN) computing. As a distributed computing paradigm, edge offloading that migrates complex tasks from IoT devices to edge-cloud servers can break through the resource limitation of IoT devices, reduce the computing burden and improve the efficiency of task processing. However, the problem of optimal offloading decision-making is NP-hard, traditional optimization methods are difficult to achieve results efficiently. Besides, there are still some shortcomings in existing deep learning methods, e.g., the slow learning speed and the failure of the original network parameters when the environment changes. To tackle these challenges, we propose a Deep Meta Reinforcement Learning-based offloading (DMRO) algorithm, which combines multiple parallel DNNs with Q-learning to make fine-grained offloading decisions. By aggregating the perceptive ability of deep learning, the decision-making ability of reinforcement learning, and the rapid environment learning ability of meta-learning, it is possible to quickly and flexibly obtain the optimal offloading strategy from the IoT environment. Simulation results demonstrate that the proposed algorithm achieves obvious improvement over the Deep Q-Learning algorithm and has strong portability in making real-time offloading decisions even in time-varying IoT environments.