Cloud databases are now foundational to enterprise digital operations; however, their elasticity and distributed architecture increase the complexity of manual administration. This review investigates how artificial intelligence can facilitate bounded autonomy in cloud database management to enhance performance, reliability, and operational efficiency, with a specific focus on Saudi enterprises. A structured integrative review synthesizes peer-reviewed literature published primarily between 2020 and 2025, spanning database systems, cloud resource management, anomaly diagnosis, autonomous tuning, and Saudi cloud adoption. Learned query optimization and cardinality estimation improve execution decisions; reinforcement and Bayesian learning accelerate configuration tuning; workload-aware resource orchestration manages latency and utilization; and AI-supported monitoring reduces diagnosis and recovery cycles. Nevertheless, the literature reveals persistent limitations: model transfer across workloads is unreliable, optimization objectives are often narrower than enterprise service-level requirements, online exploration can introduce availability risks, and most evaluations are laboratory-based rather than longitudinal production studies. Saudi adoption studies further demonstrate that security, privacy, trust, and organizational readiness influence cloud decisions, indicating that technically advanced autonomy will not be adopted without auditable controls and clear accountability. Accordingly, this review proposes a closed-loop architecture in which telemetry, AI decision engines, policy constraints, database control actions, and human oversight function as a governed feedback system. The central conclusion is that autonomous cloud database management should be implemented as progressively delegated, observable, and reversible decision-making, rather than as unrestricted automation.