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Unmanned aerial vehicle design inherently involves conflicting objectives where aerodynamic efficiency improvements often compromise structural integrity, and lightweight structures may sacrifice payload capacity or endurance. This comprehensive review examines multi-objective optimization frameworks that systematically navigate these trade-offs to generate Pareto-optimal UAV designs. We systematically analyze the formulation of UAV design as a multi-objective optimization problem, identifying primary objectives including maximizing lift-to-drag ratio, minimizing structural weight, maximizing payload fraction, extending endurance, and enhancing maneuverability, while satisfying constraints on structural strength, flutter margins, control authority, and manufacturing feasibility. The review categorizes optimization frameworks by their algorithmic approach: evolutionary multi-objective algorithms (NSGA-II, NSGA-III, MOEA/D, SPEA2), gradient-based methods with scalarization techniques (weighted sum, ε-constraint, goal programming), and surrogate-assisted optimization using Kriging, radial basis functions, or neural networks to reduce computational expense. We examine the integration of high-fidelity analysis tools within optimization loops, analyzing coupling strategies between computational fluid dynamics for aerodynamic evaluation, finite element analysis for structural assessment, and flight dynamics simulation for performance prediction. Particular attention is devoted to aeroelastic considerations where aerodynamic loads induce structural deformations that alter aerodynamic performance, requiring coupled fluid-structure interaction analysis within the optimization framework. The review synthesizes case studies spanning fixed-wing UAV configurations (conventional, flying wing, joined-wing), rotary-wing designs (single rotor, coaxial, quadrotor), and hybrid VTOL concepts, identifying common trade-off patterns and design principles. We analyze design variable parameterization approaches from low-dimensional representations using airfoil selection and planform parameters to high-dimensional free-form deformation and topology optimization enabling radical design exploration. The paper examines uncertainty quantification and robust optimization methods that account for manufacturing tolerances, operational condition variations, and model uncertainties, ensuring designs perform reliably across their operational envelope. Advanced techniques are reviewed including multi-fidelity optimization leveraging both low-cost approximate models and expensive high-fidelity simulations, adaptive sampling strategies that focus computational resources on promising design regions, and preference articulation methods incorporating designer knowledge. We critically assess visualization and decision-making tools for navigating high-dimensional Pareto fronts, including parallel coordinates, self-organizing maps, and interactive evolutionary computation. The review identifies disciplinary coupling challenges where aerodynamic optimization suggests thin, flexible wings while structural requirements demand thickness and rigidity, examining resolution strategies through morphing structures and adaptive materials. Application-specific considerations are discussed for different UAV missions: high-altitude long-endurance platforms prioritizing aerodynamic efficiency, tactical UAVs balancing agility and endurance, and cargo UAVs maximizing payload capacity.