The quantity and heterogeneity of intelligent energy generation and consumption terminals in the smart grid are increasing drastically over the years. These edge devices have created significant pressures on cloud computing (CC) system and centralised control for data storage and processing in real-time operation and control. The integration of edge computing (EC) can effectively alleviate the pressure and conduct real-time processing while ensuring data security. This paper conducts an extensive review of the EC-CC computing system and its application to the smart grid, which will integrate a vast number of dispersed devices. It first comprehensively describes the relationship among CC, fog computing (FC), and EC to provide a theoretical basis for the differentiation. It then introduces the architecture of the EC-CC computing system in the smart grid, where the architecture consists of both hardware structure and software platforms, and key technologies are introduced to support functionalities. Thereafter, the application to the smart grid is discussed across the whole supply chain, including energy generation, transportation (transmission and distribution networks), and consumption. Finally, future research opportunities and challenges of EC-CC while being applied to the smart grid are outlined. This paper can inform future research and industrial exploitations of these new technologies to enable a highly efficient smart grid under decarbonisation, digitalisation, and decentralisation transitions.