Multimodal Large Language Models (MLLMs) have expanded the capabilities of traditional language models by enabling interaction through both text and images. However, ensuring the safety of these models remains a significant challenge, particularly in accurately identifying whether multimodal content is safe or unsafe-a capability we term safety awareness. In this paper, we introduce MMSafeAware, the first comprehensive multimodal safety awareness benchmark designed to evaluate MLLMs across 29 safety scenarios with 1500 carefully curated image-prompt pairs. MMSafeAware includes both unsafe and over-safety subsets to assess models abilities to correctly identify unsafe content and avoid over-sensitivity that can hinder helpfulness. Evaluating nine widely used MLLMs using MMSafeAware reveals that current models are not sufficiently safe and often overly sensitive; for example, GPT-4V misclassifies 36.1% of unsafe inputs as safe and 59.9% of benign inputs as unsafe. We further explore three methods to improve safety awareness-prompting-based approaches, visual contrastive decoding, and vision-centric reasoning fine-tuning-but find that none achieve satisfactory performance. Our findings highlight the profound challenges in developing MLLMs with robust safety awareness, underscoring the need for further research in this area. All the code and data will be publicly available to facilitate future research.