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Although semantic communications have exhibited satisfactory performance for a large number of tasks, the impact of semantic noise and the robustness of the systems have not been well investigated. Semantic noise refers to the misleading between the intended semantic symbols and received ones, thus cause the failure of tasks. In this paper, we first propose a framework for the robust end-to-end semantic communication systems to combat the semantic noise. In particular, we analyze sample-dependent and sample-independent semantic noise. To combat the semantic noise, the adversarial training with weight perturbation is developed to incorporate the samples with semantic noise in the training dataset. Then, we propose to mask a portion of the input, where the semantic noise appears frequently, and design the masked vector quantized-variational autoencoder (VQ-VAE) with the noise-related masking strategy. We use a discrete codebook shared by the transmitter and the receiver for encoded feature representation. To further improve the system robustness, we develop a feature importance module (FIM) to suppress the noise-related and task-unrelated features. Thus, the transmitter simply needs to transmit the indices of these important task-related features in the codebook. Simulation results show that the proposed method can be applied in many downstream tasks and significantly improve the robustness against semantic noise with remarkable reduction on the transmission overhead.