The ability to acknowledge the inevitable uncertainty in their knowledge and reasoning is a prerequisite for AI systems to be truly truthful and reliable. In this paper, we present a taxonomy of uncertainty specific to vision-language AI systems, distinguishing between epistemic uncertainty (arising from a lack of information) and aleatoric uncertainty (due to inherent unpredictability), and further explore finer categories within. Based on this taxonomy, we synthesize a benchmark dataset, CertainlyUncertain, featuring 178K visual question answering (VQA) samples as contrastive pairs. This is achieved by 1) inpainting images to make previously answerable questions into unanswerable ones; and 2) using image captions to prompt large language models for both answerable and unanswerable questions. Additionally, we introduce a new metric confidence-weighted accuracy, that is well correlated with both accuracy and calibration error, to address the shortcomings of existing metrics. Despite the recent rapid progress in vision-language models (VLMs), evaluations on our benchmark show that they perform poorly in uncertain scenarios. Further experiments demonstrate that supervised fine-tuning with CertainlyUncertain enhances the performance of VLMs, and reduces the calibration error. These improvements extend beyond our benchmark to existing refusal-oriented datasets and show positive results on reducing hallucinations, while maintaining performance on standard VQA benchmarks. Our work underscores the importance of addressing uncertainty in vision-language AI systems to improve their reliability and trustworthiness in real-world applications.