Scene text segmentation aims at cropping texts from scene images, which is usually used to help generative models edit or remove texts. The existing text segmentation methods tend to involve various text-related supervisions for better performance. However, most of them ignore the importance of text edges, which are significant for downstream applications. In this paper, we propose Edge-Aware Transformers, termed EAFormer, to segment texts more accurately, especially at the edge of texts. Specifically, we first design a text edge extractor to detect edges and filter out edges of non-text areas. Then, we propose an edge-guided encoder to make the model focus more on text edges. Finally, an MLP-based decoder is employed to predict text masks. We have conducted extensive experiments on commonly-used benchmarks to verify the effectiveness of EAFormer. The experimental results demonstrate that the proposed method can perform better than previous methods, especially on the segmentation of text edges. Considering that the annotations of several benchmarks (e.g., COCOTS and MLTS) are not accurate enough to fairly evaluate our methods, we have relabeled these datasets. Through experiments, we observe that our method can achieve a higher performance improvement when more accurate annotations are used for training. The code and datasets are available at https://hyangyu.github.io/EAFormer/.