Beyond Diagonal Reconfigurable Intelligent Surfaces (BD-RIS) represent a groundbreaking innovation in sixth-generation (6G) wireless networks, enabling unprecedented control over wireless propagation environments compared to conventional diagonal RIS (D-RIS). This survey provides a comprehensive analysis of BD-RIS, detailing its architectures, operational principles, and mathematical modeling while highlighting its performance benefits. BD-RIS classifications, including single-connected, fully-connected, and group-connected architectures, and their reflective, transmissive, hybrid, and multi-sector operating modes are examined. Recent advances in BD-RIS-enabled 6G networks are reviewed, focusing on critical areas such as channel estimation, sum-rate and spectral efficiency optimization, energy efficiency enhancement, and security. The survey identifies fundamental challenges in BD-RIS research, including hardware design limitations, adaptive channel estimation, and the impact of non-ideal hardware effects. Future research directions for BD-RIS are proposed, emphasizing the integration of artificial intelligence and machine learning (AI/ML), joint optimization of communication and sensing, and enhanced physical layer security (PLS). This study concludes by underscoring BD-RIS's transformative potential to redefine 6G wireless networks, offering valuable insights and lessons for future research and development.