Unit commitment (UC) remains a central short-term scheduling problem, but high penetration of renewable energy sources (RES) has increased its uncertainty, flexibility, and security requirements. This review synthesizes UC formulations and solution methods for renewable-rich power systems, emphasizing the relationship between formulation complexity, operational requirements, and solution-method suitability. It first organizes deterministic, network-constrained, security-constrained, stochastic, robust, clustered, storage-integrated, and multi-energy UC models, highlighting how forecast errors, net-load ramps, reserve needs, curtailment, low inertia, electric-vehicle flexibility, and network congestion shape commitment decisions. It then critically assesses classical mathematical programming, decomposition, and meta heuristic methods in terms of scalability, feasibility guarantees, optimality evidence, reproducibility, and applicability to RES-integrated UC. In addition, the review classifies machine learning applications according to their primary role in the UC workflow: input and uncertainty learning, learning-assisted optimization, and assurance, validation, and deployment. Recent directions such as renewable scenario generation, surrogate modeling, warm-starting, constraint screening, graph-based learning, reinforcement learning, and decision support are examined to clarify how ML can support, rather than replace, optimization-based scheduling. The review also discusses barriers to practical deployment, including generalization to unseen operating conditions, feasibility violations, interpretability, operator trust, and lack of standardized benchmarks. Finally, it identifies hybrid ML–optimization frameworks, benchmark-driven evaluation, and validation-oriented deployment as key research needs for reliable, scalable, and secure UC in low-carbon power systems.