Achieving precise prediction and intelligent control remains a pivotal challenge in cutting processes. This need is addressed through a comprehensive survey of three critical enabling technologies: cutting force/temperature modeling, tool path planning, and vibration suppression. First, the evolution of cutting force and temperature modeling is analyzed, tracing its progression from traditional analytical methods and finite-element numerical simulations to data-driven models such as machine learning (ML) and physics-informed neural networks. This analysis highlights multiphysics coupling and model–data fusion as key to enhancing prediction accuracy. Subsequently, the evolution of tool path planning is examined, showing its development from a geometric interpolation problem into a multi-objective optimization challenge incorporating dynamic constraints, involving computational geometry, graph theory, and meta-heuristic algorithms. Finally, stability analysis based on time-delay differential equations, state identification via signal processing and ML, and active control strategies for vibration suppression are discussed. In conclusion, mathematical methods are shown to be fundamentally integrated throughout the ‘perception–prediction–decision–control’ closed-loop of the cutting process. This integration provides a solid theoretical foundation and technical support for building high-performance manufacturing systems dedicated to complex curved critical components.