In the big data era, graphs are used as effective representations of data with the complex relationship in many scenarios. Graph processing applications are widely used in various fields to dig out the potential value of graph data. The irregular execution pattern of graph processing applications introduces irregular workload, intensive read-modify-write updates, irregular memory accesses, and irregular communications. Existing general architectures cannot effectively handle the above challenges. In order to overcome these challenges, a large number of graph processing accelerator designs have been proposed. They tailor the computation pipeline, memory subsystem, storage subsystem, and communication subsystem to the graph processing application. Thanks to these hardware customizations, graph processing accelerators have achieved significant improvements in performance and energy efficiency compared with the state-of-the-art software frameworks running on general architectures. In order to allow the related researchers to have a comprehensive understanding of the graph processing accelerator, this paper first classifies and summarizes customized designs of existing work based on the computer’s pyramid organization structure from top to bottom. This article then discusses the accelerator design of the emerging graph processing application (i.e., graph neural network) with specific graph neural network accelerator cases. In the end, this article discusses the future design trend of the graph processing accelerator.