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Deep graph learning (DGL) has achieved remarkable progress in both business and scientific areas ranging from finance and e-commerce to drug and advanced material discovery. Despite the progress, applying DGL to real-world applications faces a series of reliability threats including inherent noise, distribution shift, and adversarial attacks. This survey aims to provide a comprehensive review of recent advances for improving the reliability of DGL algorithms against the above threats. In contrast to prior related surveys which mainly focus on adversarial attacks and defense, our survey covers more reliability-related aspects of DGL, i.e., inherent noise and distribution shift. Additionally, we discuss the relationships among above aspects and highlight some important issues to be explored in future research.