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Knowledge graph (KG) reasoning is becoming increasingly popular in both academia and industry. Conventional KG reasoning based on symbolic logic is deterministic, with reasoning results being explainable, while modern embedding-based reasoning can deal with uncertainty and predict plausible knowledge, often with high efficiency via vector computation. A promising direction is to integrate both logic-based and embedding-based methods, with the vision to have advantages of both. It has attracted wide research attention with more and more works published in recent years. In this paper, we comprehensively survey these works, focusing on how logics and embeddings are integrated. We first briefly introduce preliminaries, then systematically categorize and discuss works of logic and embedding-aware KG reasoning from different perspectives, and finally conclude and discuss the challenges and further directions.