Integrated localization and communications (ILAC) is a key feature of the future sixth-generation (6G) network. Benefiting from wide coverage and flexible deployment, operating ILAC in low-earth-orbit (LEO) satellite systems is a promising way to provide ubiquitously flexible localization and high-capacity communication. Particularly, in terms of localization, LEO satellite systems can break through limitations of global navigation satellite systems (GNSS) and provide superior signal quality for ILAC service. Although the LEO satellite systems are crucial for future 6G networks, they also bring challenges. First, the current signal waveforms for LEO satellite systems are primarily optimized for communication purposes, which may fail to meet the requirements of ILAC in the 6G networks. Second, the high mobility of LEO satellites leads to time-varying channels and significant Doppler shifts in satellite-ground links. Conventional positioning methods, generally tailored for terrestrial signal waveforms and assuming linear time-invariant channels, will be suboptimal in this new scenario. To address these challenges, this research aims to develop novel integrated localization and communication methods tailored for 6G LEO satellite systems with time-varying channels. To this end, three work packages (WPs) are conducted. WP1 is to design advance signal waveforms for integrated localization and communication in 6G LEO satellite systems and conduct a performance analysis considering the time-varying channel. WP2 is to develop advanced LEO satellite-based localization and tracking algorithms for moving targets. WP3 is to develop adaptive resource allocation and satellite handover algorithms for ILAC with multiple LEO satellites. This project aims to develop innovative integrated localization and communications techniques for 6G LEO satellite systems, facilitating precise positioning and high-data-rate communications for moving targets.