Recently advanced non-geostationary (NGSO) satellite networks represented by large constellations and advanced payloads provide great promises for enabling high-quality Internet connectivity to any place on Earth. However, the traditional approach to satellite operations cannot address the new challenges in the NGSO satellite networks imposed by the significant increase in complexity, security, resilience, and environmental concerns. Therefore, a reliable, sustainable, and efficient approach is required for the entire life-cycle of satellite network operations. This paper provides a timely response to the new challenges and proposes a novel approach called "SatAIOps" as an overall solution. Through our discussion on the current challenges of the advanced satellite networks, SatAIOps and its functional modules in the entire life-cycle of satellites are proposed, with some example technologies given. SatAIOps provides a new perspective for addressing operational challenges with trustworthy and responsible AI technologies. It enables a new framework for evolving and collaborative efforts from research and industry communities.