Intelligence Edge Computing (IEC) is the key enabler of emerging 5G technologies networks and beyond. IEC is considered to be a promising backbone of future services and wireless communication systems in 5G integration. In addition, IEC enables various use cases and applications, including autonomous vehicles, augmented and virtual reality, big data analytic, and other customer-oriented services. Moreover, it is one of the 5G technologies that most enhanced market drivers in different fields such as customer service, healthcare, education methods, IoT in agriculture and energy sustainability. However, 5G technological improvements face many challenges such as traffic volume, privacy, security, digitization capabilities, and required latency. Therefore, 6G is considered to be promising technology for the future. To this end, compared to other surveys, this paper provides a comprehensive survey and an inclusive overview of Intelligence Edge Computing (IEC) technologies in 6G focusing on main up-to-date characteristics, challenges, potential use cases and market drivers. Furthermore, we summarize research efforts on IEC in 5G from 2014 to 2021, in which the integration of IEC and 5G technologies are highlighted. Finally, open research challenges and new future directions in IEC with 6G networks will be discussed.