This study provides a comprehensive examination of advanced optimization techniques for the aerodynamic design of turbomachinery. Optimization methods, including adjoint and gradient-based approaches, metaheuristic-based methods, surrogate-based models, neural networks, deep learning-based techniques, and hybrid approaches are evaluated. Special attention is given to emerging machine learning applications, particularly deep learning and artificial neural networks, which are reshaping the optimization landscape. The study also highlights the challenges and limitations of these methods, addressing computational efficiency and industrial applicability. Through the analysis of recent case studies and applications, insights into future research directions and potential developments in turbomachinery optimization are provided.