The integration of satellite and terrestrial networks has emerged as a promising solution to provide seamless and ubiquitous connectivity, especially in remote and underserved areas. One of the key technologies enabling this integration is network slicing, which allows the allocation of virtualized network resources to different services and applications. This article provides a comprehensive analysis of network slicing in integrated satellite-terrestrial networks, exploring the challenges and opportunities it presents. Network slicing, a cutting edge technology, plays a pivotal role in realizing this vision by enabling the dynamic allocation of virtualized network resources to support a wide array of services and applications. We delve into the intricate landscape of technical hurdles, managerial intricacies, and regulatory complexities associated with network slicing. Additionally, we underscore the immense potential of network slicing in terms of optimizing network efficiency, accommodating a plethora of applications, and enriching the user experience. Furthermore, we outline a visionary roadmap for future research and development efforts that will shape the landscape of network slicing within integrated networks.