The development of Long-CoT reasoning has advanced LLM performance across various tasks, including language understanding, complex problem solving, and code generation. This paradigm enables models to generate intermediate reasoning steps, thereby improving both accuracy and interpretability. However, despite these advancements, a comprehensive understanding of how CoT-based reasoning affects the trustworthiness of language models remains underdeveloped. In this paper, we survey recent work on reasoning models and CoT techniques, focusing on five core dimensions of trustworthy reasoning: truthfulness, safety, robustness, fairness, and privacy. For each aspect, we provide a clear and structured overview of recent studies in chronological order, along with detailed analyses of their methodologies, findings, and limitations. Future research directions are also appended at the end for reference and discussion. Overall, while reasoning techniques hold promise for enhancing model trustworthiness through hallucination mitigation, harmful content detection, and robustness improvement, cutting-edge reasoning models themselves often suffer from comparable or even greater vulnerabilities in safety, robustness, and privacy. By synthesizing these insights, we hope this work serves as a valuable and timely resource for the AI safety community to stay informed on the latest progress in reasoning trustworthiness. A full list of related papers can be found at \href{https://github.com/ybwang119/Awesome-reasoning-safety}{https://github.com/ybwang119/Awesome-reasoning-safety}.