In this talk, we introduce our recent work on developing high-order h-adaptive finite volume methods for steady Euler equations in the complex domain. Numerical challenges such as high order representation of the curved boundary, high order reconstruction of the numerical solution, would be introduced, and our solutions for these challenges will also be provided. In addition, a goal-oriented adaptive mesh method is introduced for improving the performance of the method. Numerical results successfully show the high-order accuracy and the robustness of the proposed method.