This paper focuses on graph metric learning. First, we present a class of maximum mean discrepancy (MMD) based graph kernels, called MMD-GK. These kernels are computed by applying MMD to the node representations of two graphs with message-passing propagation. Compared to classical graph kernels such as the Weisfeiler-Lehman kernel, our MMD-GKs have much lower computational costs and are able to exploit nodes' features of graphs effectively. Secondly, we provide a class of deep MMD-GKs that are able to learn graph metrics and implicit graph features adaptively in an unsupervised manner. Thirdly, we propose a class of supervised deep MMD-GKs that are able to utilize label information of graphs and hence yield more discriminative metrics. Besides the algorithms, we provide theoretical analysis for the proposed methods. The proposed methods are evaluated in comparison to many baselines such as graph kernels and graph neural networks in the tasks of graph clustering and graph classification. The numerical results demonstrate the effectiveness and superiority of our methods.