Audio-visual representation learning aims to develop systems with human-like perception by utilizing correlation between auditory and visual information. However, current models often focus on a limited set of tasks, and generalization abilities of learned representations are unclear. To this end, we propose the AV-SUPERB benchmark that enables general-purpose evaluation of unimodal audio/visual and bimodal fusion representations on 7 datasets covering 5 audio-visual tasks in speech and audio processing. We evaluate 5 recent self-supervised models and show that none of these models generalize to all tasks, emphasizing the need for future study on improving universal model performance. In addition, we show that representations may be improved with intermediate-task fine-tuning and audio event classification with AudioSet serves as a strong intermediate task. We release our benchmark with evaluation code and a model submission platform to encourage further research in audio-visual learning.