Satellite-Terrestrial Network (STN) is one of the core of future communication technology and has been in the spotlight in recent years by various academia and industries. Above all, Low-Earth Orbit (LEO) satellite technology is attracting attention since it can exhibit high energy efficiency among satellites in various orbits, and related business projects are also being carried out on a large scale. The rapid emergence of advanced mobile devices and applications in recent years has made communication technologies demand better quality of service (QoS). To meet these needs, Mobile Edge Computing (MEC) technology has been studied a lot, and now many studies that applied MEC with STN which is called Satellite Mobile Edge Computing (SMEC) have started to be conducted. However, SMEC has a number of considerations differently from terrestrial networks due to the characteristics of satellite communication. Most current SMEC studies focus on optimizing energy consumption of STN system, which simply apply the MEC technique of the existing terrestrial network to STN as it is. It seems to be unrealistic since the satellite communication environment has not been considered. Therefore, in this study, comprehensive concepts and cases are presented through a wide survey on STN and SMEC. Additionally, a simple and realistic SMEC optimization scenario considering the environment of satellite communication is proposed.