Recent advances in Omni-Multimodal Large Language Models (Omni-MLLMs) have enabled strong integration of vision, audio, and language. However, their audio-visual intelligence (AVI) remains insufficiently evaluated due to the lack of systematic and comprehensive benchmarks. We introduce AVI-Bench, a cognitively inspired benchmark that evaluates Omni-MLLMs across three stages, perception, understanding, and reasoning, through cross-modal tasks requiring joint audio-visual interpretation. This design enables fine-grained diagnosis of model capabilities and failure modes. To further assess robustness beyond familiar domains, we propose AVI-Bench-PriSe, an extension that probes models' primitive audio-visual sensation using unfamiliar, low-semantic stimuli, testing generalization beyond common training distributions. Extensive experiments on both open-source and closed-source models reveal substantial limitations in current Omni-MLLMs. Based on these findings, we present a four-level AVI taxonomy. Overall, AVI-Bench provides a principled evaluation framework to guide the development of more robust and generalizable AVI. Project website: https://fudancvl.github.io/AVI-Bench/