Audio-visual speech contains synchronized audio and visual information that provides cross-modal supervision to learn representations for both automatic speech recognition (ASR) and visual speech recognition (VSR). We introduce continuous pseudo-labeling for audio-visual speech recognition (AV-CPL), a semi-supervised method to train an audio-visual speech recognition (AVSR) model on a combination of labeled and unlabeled videos with continuously regenerated pseudo-labels. Our models are trained for speech recognition from audio-visual inputs and can perform speech recognition using both audio and visual modalities, or only one modality. Our method uses the same audio-visual model for both supervised training and pseudo-label generation, mitigating the need for external speech recognition models to generate pseudo-labels. AV-CPL obtains significant improvements in VSR performance on the LRS3 dataset while maintaining practical ASR and AVSR performance. Finally, using visual-only speech data, our method is able to leverage unlabeled visual speech to improve VSR.