Deep Learning models based on heatmap regression have revolutionized the task of facial landmark localization with existing models working robustly under large poses, non-uniform illumination and shadows, occlusions and self-occlusions, low resolution and blur. However, despite their wide adoption, heatmap regression approaches suffer from discretization-induced errors related to both the heatmap encoding and decoding process. In this work we show that these errors have a surprisingly large negative impact on facial alignment accuracy. To alleviate this problem, we propose a new approach for the heatmap encoding and decoding process by leveraging the underlying continuous distribution. To take full advantage of the newly proposed encoding-decoding mechanism, we also introduce a Siamese-based training that enforces heatmap consistency across various geometric image transformations. Our approach offers noticeable gains across multiple datasets setting a new state-of-the-art result in facial landmark localization. Code alongside the pretrained models will be made available at https://www.adrianbulat.com/face-alignment