Graph neural networks (GNNs) are powerful deep learning models for graph-structured data, demonstrating remarkable success across diverse domains. Recently, the database (DB) community has increasingly recognized the potentiality of GNNs, prompting a surge of researches focusing on improving database systems through GNN-based approaches. However, despite notable advances, There is a lack of a comprehensive review and understanding of how GNNs could improve DB systems. Therefore, this survey aims to bridge this gap by providing a structured and in-depth overview of GNNs for DB systems. Specifically, we propose a new taxonomy that classifies existing methods into two key categories: (1) Relational Databases, which includes tasks like performance prediction, query optimization, and text-to-SQL, and (2) Graph Databases, addressing challenges like efficient graph query processing and graph similarity computation. We systematically review key methods in each category, highlighting their contributions and practical implications. Finally, we suggest promising avenues for integrating GNNs into Database systems.