Medical image segmentation plays a vital role in computer-aided diagnosis (CAD) systems. Both convolutional neural networks (CNNs) with strong local information extraction capacities and transformers with excellent global representation capacities have achieved remarkable performance in medical image segmentation. However, because of the semantic differences between local and global features, how to combine convolution and transformers effectively is an important challenge in medical image segmentation. In this paper, we proposed TransConver, a U-shaped segmentation network based on convolution and transformer for automatic and accurate brain tumor segmentation in MRI images. Unlike the recently proposed transformer and convolution based models, we proposed a parallel module named transformer-convolution inception (TC-inception), which extracts local and global information via convolution blocks and transformer blocks, respectively, and integrates them by a cross-attention fusion with global and local feature (CAFGL) mechanism. Meanwhile, the improved skip connection structure named skip connection with cross-attention fusion (SCCAF) mechanism can alleviate the semantic differences between encoder features and decoder features for better feature fusion. In addition, we designed 2D-TransConver and 3D-TransConver for 2D and 3D brain tumor segmentation tasks, respectively, and verified the performance and advantage of our model through brain tumor datasets. We trained our model on 335 cases from the training dataset of MICCAI BraTS2019 and evaluated the model's performance based on 66 cases from MICCAI BraTS2018 and 125 cases from MICCAI BraTS2019. Our TransConver achieved the best average Dice score of 83.72% and 86.32% on BraTS2019 and BraTS2018, respectively. We proposed a transformer and convolution parallel network named TransConver for brain tumor segmentation. The TC-Inception module effectively extracts global information while retaining local details. The experimental results demonstrated that good segmentation requires the model to extract local fine-grained details and global semantic information simultaneously, and our TransConver effectively improves the accuracy of brain tumor segmentation.