Graph neural networks (GNNs) have gained significant popularity for classification tasks in machine learning, yet their applications to regression problems remain limited. Concurrently, attention mechanisms have emerged as powerful tools in sequential learning tasks. In this paper, we employ GNNs and attention mechanisms to address a classical but challenging nonlinear regression problem: network localization. We propose a novel GNN-based network localization method that achieves exceptional stability and accuracy in the presence of severe non-line-of-sight (NLOS) propagations, while eliminating the need for laborious offline calibration or NLOS identification. Extensive experimental results validate the effectiveness and high accuracy of our GNN-based localization model, particularly in challenging NLOS scenarios. However, the proposed GNN-based model exhibits limited flexibility, and its accuracy is highly sensitive to a specific hyperparameter that determines the graph structure. To address the limitations and extend the applicability of the GNN-based model to real scenarios, we introduce two attentional graph neural networks (AGNNs) that offer enhanced flexibility and the ability to automatically learn the optimal hyperparameter for each node. Experimental results confirm that the AGNN models are able to enhance localization accuracy, providing a promising solution for real-world applications. We also provide some analyses of the improved performance achieved by the AGNN models from the perspectives of dynamic attention and signal denoising characteristics.

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