This paper presents a survey of the theoretical development of semantic communication, focusing on semantic information theory. Semantic communication aims to efficiently exchange semantics rather than exact data, emphasizing the accurate conveyance of meaning. The paper introduces key concepts in semantic information theory, including semantic entropy, semantic rate-distortion, and semantic channel capacity. It discusses the challenges and open problems in semantic communication, such as the measurement of semantic information, the design of semantic coding, and the integration of semantic information theory with Shannon's information theory. The paper also reviews various mathematical theories and tools, including information bottleneck and age of information, and evaluates their applicability in semantic communication. It highlights the importance of semantic communication in emerging intelligent services, where ultra-low latency and high-throughput capabilities are essential. The paper emphasizes the need for a comprehensive theoretical framework to address the challenges in semantic communication, including the fundamental limits of semantic communication, the capabilities of semantic-aware networks, and the application of theoretical guidance for deep learning in semantic communication. The paper concludes with a discussion of the potential challenges and future directions in semantic communication and semantic information theory.This paper presents a survey of the theoretical development of semantic communication, focusing on semantic information theory. Semantic communication aims to efficiently exchange semantics rather than exact data, emphasizing the accurate conveyance of meaning. The paper introduces key concepts in semantic information theory, including semantic entropy, semantic rate-distortion, and semantic channel capacity. It discusses the challenges and open problems in semantic communication, such as the measurement of semantic information, the design of semantic coding, and the integration of semantic information theory with Shannon's information theory. The paper also reviews various mathematical theories and tools, including information bottleneck and age of information, and evaluates their applicability in semantic communication. It highlights the importance of semantic communication in emerging intelligent services, where ultra-low latency and high-throughput capabilities are essential. The paper emphasizes the need for a comprehensive theoretical framework to address the challenges in semantic communication, including the fundamental limits of semantic communication, the capabilities of semantic-aware networks, and the application of theoretical guidance for deep learning in semantic communication. The paper concludes with a discussion of the potential challenges and future directions in semantic communication and semantic information theory.