Documentation plays a crucial role in both external accountability and internal governance of AI systems. Although there are many proposals for documenting AI data, models, systems, and methods, the ways these practices enhance governance as well as the challenges practitioners and organizations face with documentation remain underexplored. In this paper, we analyze 37 proposed documentation frameworks and 22 empirical studies evaluating their use. We identify several pathways or "theories of change" through which documentation can enhance governance, including informing stakeholders about AI risks and applications, facilitating collaboration, encouraging ethical deliberation, and supporting best practices. However, empirical findings reveal significant challenges for practitioners, such as insufficient incentives and resources, structural and organizational communication barriers, interpersonal and organizational constraints to ethical action, and poor integration with existing workflows. These challenges often hinder the realization of the possible benefits of documentation. We also highlight key considerations for organizations when designing documentation, such as determining the appropriate level of detail and balancing automation in the process. We conclude by discussing how future research can expand on our findings such as by exploring documentation approaches that support governance of general-purpose models and how multiple transparency and documentation methods can collectively improve governance outcomes.