Recommender system is one of the most important information services on today's Internet. Recently, graph neural networks have become the new state-of-the-art approach to recommender systems. In this survey, we conduct a comprehensive review of the literature on graph neural network-based recommender systems. We first introduce the background and the history of the development of both recommender systems and graph neural networks. For recommender systems, in general, there are four aspects for categorizing existing works: stage, scenario, objective, and application. For graph neural networks, the existing methods consist of two categories, spectral models and spatial ones. We then discuss the motivation of applying graph neural networks into recommender systems, mainly consisting of the high-order connectivity, the structural property of data, and the enhanced supervision signal. We then systematically analyze the challenges in graph construction, embedding propagation/aggregation, model optimization, and computation efficiency. Afterward and primarily, we provide a comprehensive overview of a multitude of existing works of graph neural network-based recommender systems, following the taxonomy above. Finally, we raise discussions on the open problems and promising future directions in this area. We summarize the representative papers along with their code repositories in \url{https://github.com/tsinghua-fib-lab/GNN-Recommender-Systems}.