In this article, unconstrained trust region methods are studied and applied in aerodynamic shape optimization. Trust region methods are high-performance gradient-based optimization methods. The theory of unconstrained trust region methods is explained, and a trust region subproblem is solved using the dogleg method and the More-Sorensen method. After verifying the trust region methods via analytical test problems, they are applied to the aerodynamic shape design optimization for transonic and subsonic flows.