The rapid development of new information and communication technologies (ICTs) and the deployment of advanced Internet of Things (IoT)-based devices has led to the study and implementation of edge computing technologies in smart grid (SG) systems. In addition, substantial work has been expended in the literature to incorporate artificial intelligence (AI) techniques into edge computing, resulting in the promising concept of edge intelligence (EI). Consequently, in this article, we provide an overview of the current state-of-the-art in terms of EI-based SG adoption from a range of angles, including architectures, computation offloading, and cybersecurity concerns. The basic objectives of this article are fourfold. To begin, we discuss EI and SGs separately. Then we highlight contemporary concepts closely related to edge computing, fundamental characteristics, and essential enabling technologies from an EI perspective. Additionally, we discuss how the use of AI has aided in optimizing the performance of edge computing. We have emphasized the important enabling technologies and applications of SGs from the perspective of EI-based SGs. Second, we explore both general edge computing and architectures based on EI from the perspective of SGs. Thirdly, two basic questions about computation offloading are discussed: what is computation offloading and why do we need it? Additionally, we divided the primary articles into two categories based on the number of users included in the model, either a single user or a multiple user instance. Finally, we review the cybersecurity threats with edge computing and the methods used to mitigate them in SGs. Therefore, this survey comes to the conclusion that most of the viable architectures for EI in smart grids often consist of three layers: device, edge, and cloud. In addition, it is crucial that computation offloading techniques must be framed as optimization problems and addressed effectively in order to increase system performance. This article typically intends to serve as a primer for emerging and interested scholars concerned with the study of EI in SGs.