Semantic Edge Computing (SEC) and Semantic Communications (SemComs) have been proposed as viable approaches to achieve real-time edge-enabled intelligence in sixth-generation (6G) wireless networks. On one hand, SemCom leverages the strength of Deep Neural Networks (DNNs) to encode and communicate the semantic information only, while making it robust to channel distortions by compensating for wireless effects. Ultimately, this leads to an improvement in the communication efficiency. On the other hand, SEC has leveraged distributed DNNs to divide the computation of a DNN across different devices based on their computational and networking constraints. Although significant progress has been made in both fields, the literature lacks a systematic view to connect both fields. In this work, we fulfill the current gap by unifying the SEC and SemCom fields. We summarize the research problems in these two fields and provide a comprehensive review of the state of the art with a focus on their technical strengths and challenges.