Graph neural networks (GNNs) establish a deep learning framework for non-Euclidean spatial data. Compared with traditional network embedding methods, they perform deeper aggregating operations on graph structures. In recent years, GNNs have been extended to complex graphs. Nevertheless, there lacks qualified surveys which give comprehensive and systematic classification and summary on GNNs based on complex graphs. This study divides the complex graphs into 3 categories, i.e., heterogeneous graphs, dynamic graphs, and hypergraphs. GNNs based on heterogeneous graphs are divided into 2 types, i.e., edge-type aware and meta-path aware, according to the procedure that the information is aggregated. Dynamic GNNs graphs are divided into three categories: RNN-based methods, autoencoderbased methods, and spatio-temporal graph neural networks. Hypergraph GNNs are divided into expansion methods and non-expansion methods, and the expansion methods are further divided into star-expansion, clique-expansion, and line-expansion according to the expansion mode they use. The core idea of every method is illustrated in detail, the advantages and disadvantages of different algorithms are compared, the key procedures, (cross) application fields, and commonly used data sets of different complex graph GNNs are systematically listed, and some possible research directions are proposed.