With the accelerated popularization of 5G applications, accurate cellular traffic prediction is becoming increasingly important for efficient network management. Currently, the latest algorithms for cellular traffic prediction generally neglect extraction of the shallow features of cellular traffic and the prediction accuracy is hence limited. Therefore, we propose a global-local spatial-temporal transformer network (GLSTTN) that can fully excavate diverse spatial-temporal characteristics of cellular traffic for accurate cellular traffic prediction. Specifically, GLSTTN achieves this goal by constructing two modules: the global spatial-temporal module and the local spatial-temporal module. In the global spatial-temporal module, GLSTTN captures global correlations using stacked spatial-temporal blocks, where each block is composed of one spatial transformer and one temporal transformer. A skip connection is then used in each block to strengthen feature propagation. In the local spatial-temporal module, GLSTTN fully extracts the local spatial-temporal dependencies hidden in globally encoded features using densely connected convolutional neural networks. Extensive experiments demonstrate that GLSTTN achieves more accurate cellular traffic prediction than existing approaches on a real-world cellular traffic dataset.