Triangle mesh segmentation is an important task in 3D shape analysis, especially in applications such as digital humans and AR/VR. Transformer model is inherently permutation-invariant to input, which makes it a suitable candidate model for 3D mesh processing. However, two main challenges involved in adapting Transformer from natural languages to 3D mesh are yet to be solved, such as i) extracting the multi-scale information of mesh data in an adaptive manner; ii) capturing geometric structures of mesh data as the discriminative characteristics of the shape. Current point based Transformer models fail to tackle such challenges and thus provide inferior performance for discretized surface segmentation. In this work, heat diffusion based method is exploited to tackle these problems. A novel Transformer model called MeshFormer is proposed, which i) integrates Heat Diffusion method into Multi-head Self-Attention operation (HDMSA) to adaptively capture the features from local neighborhood to global contexts; ii) applies a novel Heat Kernel Signature based Structure Encoding (HKSSE) to embed the intrinsic geometric structures of mesh instances into Transformer for structure-aware processing. Extensive experiments on triangle mesh segmentation validate the effectiveness of the proposed MeshFormer model and show significant improvements over current state-of-the-art methods.