How does velodrome work?

Could you elaborate on the mechanics and operations of a velodrome? Specifically, I'm curious about the track design, the cycling technique involved, and the safety measures that are taken to ensure riders' well-being. How do the banking angles of the track influence the cyclists' speed and stability? Are there specific rules and regulations governing the use of the velodrome? And lastly, what are some of the key challenges that cyclists face when competing on such a track? I'm interested in understanding the entire process from a technical and competitive standpoint.