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Recently, heatmap regression has been widely explored in facial landmark detection and obtained remarkable performance. However, most of the existing heatmap regression-based facial landmark detection methods neglect to explore the high-order feature correlations, which is very important to learn more representative features and enhance shape constraints. Moreover, no explicit global shape constraints have been added to the final predicted landmarks, which leads to a reduction in accuracy. To address these issues, in this paper, we propose a Multi-order Multi-constraint Deep Network (MMDN) for more powerful feature correlations and shape constraints learning. Specifically, an Implicit Multi-order Correlating Geometry-aware (IMCG) model is proposed to introduce the multi-order spatial correlations and multi-order channel correlations for more discriminative representations. Furthermore, an Explicit Probability-based Boundary-adaptive Regression (EPBR) method is developed to enhance the global shape constraints and further search the semantically consistent landmarks in the predicted boundary for robust facial landmark detection. It's interesting to show that the proposed MMDN can generate more accurate boundary-adaptive landmark heatmaps and effectively enhance shape constraints to the predicted landmarks for faces with large pose variations and heavy occlusions. Experimental results on challenging benchmark datasets demonstrate the superiority of our MMDN over state-of-the-art facial landmark detection methods. The code has been publicly available at https://github.com/junwan2014/MMDN-master.