Graph neural networks (GNNs), which extend traditional neural networks for processing graph-structured data, have been widely used in many fields. The GNN computation mainly consists of the edge processing to generate messages by combining the edge/vertex features and the vertex processing to update the vertex features with aggregated messages. In addition to nontrivial vector operations in the edge processing, huge random accesses and neural network operations in the vertex processing, the graph topology of GNNs may also vary during the computation (i.e., dynamic GNNs). The above characteristics pose significant challenges on existing architectures. In this article, we propose a novel accelerator named CAMBRICON-G for efficient processing of both dynamic and static GNNs. The key of CAMBRICON-G is to abstract the irregular computation of a broad range of GNN variants to the process of regularly tiled adjacent cuboid (which extends the traditional adjacent matrix of graph by adding the dimension of vertex features). The intuition is that the adjacent cuboid facilitates exploitation of both data locality and parallelism by offering multidimensional multilevel tiling (including spatial and temporal tiling) opportunities. To perform the multidimensional spatial tiling , the CAMBRICON-G architecture mainly consists of the cuboid engine (CE) and hybrid on-chip memory. The CE has multiple vertex processing units (VPUs) working in a coordinated manner to efficiently process the sparse data and dynamically update the graph topology with dedicated instructions. The hybrid on-chip memory contains the topology-aware cache and multiple scratchpad memory to reduce off-chip memory access. To perform the multidimensional temporal tiling , an easy-to-use programming model is provided to flexibly explore different tiling options for large graphs. Experimental results show that compared against Nvidia P100 GPU, the performance and energy efficiency can be improved by and , respectively, on various GNNs, which validates both the versatility and energy efficiency of CAMBRICON-G.