Title: LipsFormer: Introducing Lipschitz Continuity to Vision Transformers. Abstract: We present a Lipschitz continuous Transformer, called LipsFormer, to pursue training stability both theoretically and empirically for Transformer-based models. In contrast to previous practical tricks that address training instability by learning rate warmup, layer normalization, attention formulation, and weight initialization, we show that Lipschitz continuity is a more essential property to ensure training stability. In LipsFormer, we replace unstable Transformer component modules with Lipschitz continuous counterparts:  CenterNorm instead of LayerNorm, spectral initialization instead of Xavier initialization, scaled cosine similarity attention instead of dot-product attention, and weighted residual shortcut. We prove that these introduced modules are Lipschitz continuous and derive an upper bound on the Lipschitz constant of LipsFormer. Our experiments show that LipsFormer allows stable training of deep Transformer architectures without the need of careful learning rate tuning such as warmup, yielding a faster convergence and better generalization. As a result,  on the ImageNet 1K dataset, LipsFormer-Tiny training for 100 epochs without learning rate warmup attains a top-1 accuracy of 81.6\% which is higher than Swin Transformer-Tiny training for 300 epochs with warmup. Moreover, LipsFormer-Tiny training for 300 epochs achieves a top-1 accuracy of 83.5\% with 4.7G FLOPs and 24M parameters. 