The integration of satellite and terrestrial networks have become inevitable in the next generations of communications networks due to emerging needs of ubiquitous connectivity of remote locations. New and existing services and critical infrastructures in remote locations in sea, land and space will be seamlessly connected through a diverse set of terrestrial and non-terrestrial communication technologies. However, the integration of terrestrial and non-terrestrial systems will open both systems to unique security challenges that can arise due to the migration of security challenges from one to another. Similarly, security challenges can also arise due to the incompatibility of distinct systems or incoherence of security policies. The resulting security implications, thus, can be highly consequential due to the criticality of the infrastructures such as space stations, autonomous ships, and aeroplanes, for instance. Therefore, in this article we study existing security challenges in satellite-terrestrial communication systems and discuss potential solutions for those challenges. Furthermore, we provide important research directions to stir future research on existing security gaps in this direction.