Optimization plays a pivotal role in aerodynamic design, enabling the improvement of performance metrics such as lift-to-drag ratio, stability, and fuel efficiency. This paper reviews key mathematical methods employed in aerodynamic optimization, including gradient-based techniques, evolutionary algorithms, surrogate modeling, and adjoint methods. The integration of computational fluid dynamics (CFD) with optimization algorithms is emphasized to address complex design problems with multiple objectives and constraints. Applications range from aircraft wing design to turbine blade shaping, demonstrating how mathematical optimization accelerates innovation in aerospace engineering. Future trends and challenges in high-fidelity modeling and real-time optimization are also discussed.