Recently, the emerging graph neural networks (GNNs) have received extensive attention from academia and industry due to the powerful graph learning and reasoning capabilities, and are considered to be the core force that promotes the field of artificial intelligence into the “cognitive intelligence” stage. Since GNNs integrate the execution process of both traditional graph processing and neural network, a hybrid execution pattern naturally exists, which makes irregular and regular computation and memory access behaviors coexist. This execution pattern makes traditional processors and the existing graph processing and neural network acceleration architectures unable to cope with the two opposing execution behaviors at the same time, and cannot meet the acceleration requirements of GNNs. To solve the above problems, acceleration architectures tailored for GNNs continue to emerge. They customize computing hardware units and on-chip storage levels for GNNs, optimize computation and memory access behaviors, and have achieved acceleration effects well. Based on the challenges faced by the GNN acceleration architectures in the design process, this paper systematically analyzes and introduces the overall structure design and the key optimization technologies in this field from computation, on-chip memory access, off-chip memory access respectively. Finally, the future direction of GNN acceleration structure design is prospected from different angles, and it is expected to bring certain inspiration to researchers in this field.