While audio-visual learning equips models with a richer understanding of the real world by leveraging multiple sensory modalities, this integration also introduces new vulnerabilities to adversarial attacks. In this paper, we present a comprehensive study of the adversarial robustness of audio-visual models, considering both temporal and modality-specific vulnerabilities. We propose two powerful adversarial attacks: 1) a temporal invariance attack that exploits the inherent temporal redundancy across consecutive time segments and 2) a modality misalignment attack that introduces incongruence between the audio and visual modalities. These attacks are designed to thoroughly assess the robustness of audio-visual models against diverse threats. Furthermore, to defend against such attacks, we introduce a novel audio-visual adversarial training framework. This framework addresses key challenges in vanilla adversarial training by incorporating efficient adversarial perturbation crafting tailored to multi-modal data and an adversarial curriculum strategy. Extensive experiments in the Kinetics-Sounds dataset demonstrate that our proposed temporal and modality-based attacks in degrading model performance can achieve state-of-the-art performance, while our adversarial training defense largely improves the adversarial robustness as well as the adversarial training efficiency.