Laser welding is widely used in complex manufacturing processes and valued for its reliability, flexibility, and high energy density. However, achieving the desired weld quality requires the detection and, ideally, the prevention of defects. Besides other methods, machine learning (ML) has been integrated into laser welding with the primary goal of process optimization and quality improvement, for example, by enabling process adaptation before or during welding to reduce defects. This survey systematically reviews publications from 2015 to 2025 that integrate machine learning and deep learning methods into laser welding optimization or adaptation processes. An extensive analysis identifies which parts of the process and for what purposes ML methods are researched and implemented and how they are evaluated, as well as the sensors, lasers, and materials involved. Furthermore, the findings are analyzed and organized into taxonomies that define overarching meta-categories into which existing approaches can be classified and contextualized. The results reveal that various ML approaches are applied for tasks, such as surrogate modeling, process planning, direct control, and virtual sensing and monitoring. Although many different control parameters and optimization targets are considered, laser power and welding speed dominate as the most frequently adjusted parameters, while penetration depth and weld geometry-related properties are the most common optimization targets. Finally, the survey identifies major challenges, including the lack of benchmarking datasets, standardized evaluation protocols, and interpretable models.