The recent performance of facial landmark detection has been significantly improved by using deep Convolutional Neural Networks (CNNs), especially the Heatmap Regression Models (HRMs). Although their performance on common benchmark datasets has reached a high level, the robustness of these models still remains a challenging problem in the practical use under noisy conditions of realistic environments. Contrary to most existing work focusing on the design of new models, we argue that improving the robustness requires rethinking many other aspects, including the use of datasets, the format of landmark annotation, the evaluation metric as well as the training and detection algorithm itself. In this paper, we propose a novel method for robust facial landmark detection, using a loss function based on the 2D Wasserstein distance combined with a new landmark coordinate sampling relying on the barycenter of the individual probability distributions. Our method can be plugged-and-play on most state-of-the-art HRMs with neither additional complexity nor structural modifications of the models. Further, with the large performance increase, we found that current evaluation metrics can no longer fully reflect the robustness of these models. Therefore, we propose several improvements to the standard evaluation protocol. Extensive experimental results on both traditional evaluation metrics and our evaluation metrics demonstrate that our approach significantly improves the robustness of state-of-the-art facial landmark detection models.