With the increasing global communication demands and the development of Internet of Things (IoT), extending the connectivity to rural and remote areas has become imperative for future networks. The sixth-generation (6G) network is expected to provide heterogeneous services and seamless network coverage for everyone and everything. Combining the advantages of both satellite and terrestrial networks, the integrated satellite-terrestrial network architecture is promising to provide global broadband access for all types of users, which has drawn much attention from both the academia and industry. In this article, we present a comprehensive survey of the state-of-the-art of integrated satellite-terrestrial networks toward 6G. First, an executive classification and summary of the integration architecture is presented from network design to performance optimization. Then, typical applications of the integrated satellite-terrestrial network are discussed based on the architecture. By considering the unique characteristics of the two networks, main challenges are pointed out when performing integration, such as the long propagation delay, complex link conditions, and high dynamics of the network topology. Finally, some promising future techniques are explored from the perspective of the integrated architecture. A detailed survey of the potential integration architectures is of great importance to enable more flexible network design and construction in future 6G networks. This article will provide a valuable guideline on future research and development of integrated satellite-terrestrial networks.