This paper investigates the capabilities of Large Language Models (LLMs) in understanding their knowledge and uncertainty over questions. Specifically, we focus on addressing known-unknown questions, characterized by high uncertainty due to the absence of definitive answers. To facilitate our study, we collect a new dataset with Known-Unknown Questions (KUQ) and establish a categorization framework to clarify the origins of uncertainty in such queries. Subsequently, we examine the performance of open-source LLMs, fine-tuned using this dataset, in distinguishing between known and unknown queries within open-ended question-answering scenarios. The fine-tuned models demonstrated a significant improvement, achieving a considerable increase in F1-score relative to their pre-fine-tuning state. Through a comprehensive analysis, we reveal insights into the models’ improved uncertainty articulation and their consequent efficacy in multi-agent debates. These findings help us understand how LLMs can be trained to identify and express uncertainty, improving our knowledge of how they understand and express complex or unclear information.