Persuasion, as one of the crucial abilities in human communication, has garnered extensive attention from researchers within the field of intelligent dialogue systems. Developing dialogue agents that can persuade others to accept certain standpoints is essential to achieving truly intelligent and anthropomorphic dialogue systems. Benefiting from the substantial progress of Large Language Models (LLMs), dialogue agents have acquired an exceptional capability in context understanding and response generation. However, as a typical and complicated cognitive psychological system, persuasive dialogue agents also require knowledge from the domain of cognitive psychology to attain a level of human-like persuasion. Consequently, the cognitive strategy-enhanced persuasive dialogue agent (defined as CogAgent), which incorporates cognitive strategies to achieve persuasive targets through conversation, has become a predominant research paradigm. To depict the research trends of CogAgent, in this paper, we first present several fundamental cognitive psychology theories and give the formalized definition of three typical cognitive strategies, including the persuasion strategy, the topic path planning strategy, and the argument structure prediction strategy. Then we propose a new system architecture by incorporating the formalized definition to lay the foundation of CogAgent. Representative works are detailed and investigated according to the combined cognitive strategy, followed by the summary of authoritative benchmarks and evaluation metrics. Finally, we summarize our insights on open issues and future directions of CogAgent for upcoming researchers.