Open source software (OSS) projects are critical to modern software systems, yet their sustainability depends on more than technical quality. OSS communities coordinate maintenance work, define authority, distribute responsibilities, onboard and retain contributors, preserve trust, and adapt to emerging organizational and technological pressures. This doctoral research investigates governance as a socio-technical foundation for OSS project health. The dissertation is organized around five studies that examine how governance shapes sustainability across multiple layers: its measurable impact on project health, its formalization in written governance artifacts, its enactment in everyday maintenance practices, its adaptation to AI-mediated participation, and its role in financial resource allocation. The completed studies show that governance reform can improve project health outcomes in an OSS community and that governance documents reveal how projects define roles, responsibilities, privileges, and authority, while also exposing ambiguity, role overlap, and responsibility accumulation. The planned studies will compare written governance with pull request review behavior, investigate how communities regulate AI-assisted and agentic participation, and examine how OSS projects govern funding, accountability, fairness, and external support. This dissertation contributes: (1) a role-authority model for understanding how OSS projects define responsibilities, privileges, and decision-making authority; (2) a governance-conformance analysis method for comparing written governance with observable maintenance behavior; and (3) empirical accounts of how OSS communities govern emerging sustainability challenges, including AI-mediated participation and financial resource allocation.