Nowadays, deep neural networks (DNNs) are the core enablers for many emerging edge AI applications. Conventional approaches for training DNNs are generally implemented at central servers or cloud centers for centralized learning, which is typically time-consuming and resource-demanding due to the transmission of a large number of data samples from the edge device to the remote cloud. To overcome these disadvantages, we consider accelerating the learning process of DNNs on the Mobile-Edge-Cloud Computing (MECC) paradigm. In this paper, we propose HierTrain, a hierarchical edge AI learning framework, which efficiently deploys the DNN training task over the hierarchical MECC architecture. We develop a novel hybrid parallelism method, which is the key to HierTrain, to adaptively assign the DNN model layers and the data samples across the three levels of the edge device, edge server and cloud center. We then formulate the problem of scheduling the DNN training tasks at both layer-granularity and sample-granularity. Solving this optimization problem enables us to achieve the minimum training time. We further implement a hardware prototype consisting of an edge device, an edge server and a cloud server, and conduct extensive experiments on it. Experimental results demonstrate that HierTrain can achieve up to 6.9× speedups compared to the cloud-based hierarchical training approach. INDEX TERMS Edge AI, deep learning, fast model training, mobile-edge-cloud computing.