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The ever-increasing demand for ubiquitous and differentiated services at anytime and anywhere emphasizes the necessity of aerospace integrated networks (AINs) which consist of multi-tiers, e.g., high-altitude platforms (HAPs) tier, unmanned aerial vehicles (UAVs) tier, low earth orbit (LEO) satellites tier, etc. for wide coverage and high capacity. However, the inherent heterogeneity, time-variability, and differentiated mobility characteristics between terrestrial networks and AINs constraint the materialization of the aforementioned widely expected potentials of AINs. Therefore, it is crucial to make a profound understanding of AINsimpacts on pivotal models and methodologies for enabling 6G, thus providing technology reference for AINs empowering 6G. To understand the latest development and ultimately open new research niches on this significant topic, this survey is the pioneer paper to serve as a systematical and comprehensive overview in both single-tier scenarios and combining-tiers scenarios in AINs. We start with a profound discussion about the state-of-the-art potentially promising methodologies and models in the era of AINs empowering 6G from the perspective of the system architecture and networking design and enabling technologies of AINs, to enable flexible and scalable management and control of the AINs and ensure network stability. Furthermore, we make an in-depth literature overview across the network dynamics modeling, theoretical performance analysis model, and system optimization to enhance the system performance and guarantee user-on-demand services. Additionally, we highlight the ongoing research challenges and future directions with focusing on the development trend of ultra-dense satellite constellations in 6G.