Quadruped robots are able to traverse most terrains on the earth using a wide variety of gaits, providing solutions for robots to operate in specialized environments. Although existing methods have achieved excellent performance in gait generation, they suffer from catastrophic forgetting and inability to evolve when encountering new gait demands. Fortunately, continual learning provides a powerful tool to address this issue and has demonstrated the effectiveness in classification, recognition, and other fields but not gait generation. This paper presents a framework of Multi-critic continual learning with experience replay (MCLER) to enable incremental gait learning. MCLER allows quadruped robots to be lifelong learners and acquire new gaits without forgetting. Extensive experiments conducted in simulated and real environments show that MCLER significantly outperforms the state-of-the-art methods. Supplementary video demonstration can be accessed via the following link: https://vsislab.github.io/mcler/.