Deep learning can provide rapid brain age estimation based on brain magnetic resonance imaging (MRI). However, most studies use one neural network to extract the global information from the whole input image, ignoring the local fine-grained details. In this paper, we propose a global-local transformer, which consists of a global-pathway to extract the global-context information from the whole input image and a local-pathway to extract the local fine-grained details from local patches. The fine-grained information from the local patches are fused with the global-context information by the attention mechanism, inspired by the transformer, to estimate the brain age. We evaluate the proposed method on 8 public datasets with 8,379 healthy brain MRIs with the age range of 0-97 years. 6 datasets are used for cross-validation and 2 datasets are used for evaluating the generality. Comparing with other state-of-the-art methods, the proposed global-local transformer reduces the mean absolute error of the estimated ages to 2.70 years and increases the correlation coefficient of the estimated age and the chronological age to 0.9853. In addition, our proposed method provides regional information of which local patches are most informative for brain age estimation. Our source code is available on: https://github.com/shengfly/global-local-transformer.