Adaptive mesh refinement is one of the most important tools in Computational Fluid Dynamics (CFD) for solving complex engineering design problems. The paper investigates two practical transonic aerofoil design optimization problems using a genetic algorithm coupled with an Euler aerodynamic analysis tool. The first problem consists in the minimization of transonic drag whereas the second is a reconstruction transonic problem solved by minimizing the pressure error. In both cases, the solutions obtained with and without adaptive mesh refinement are compared. Numerical results obtained by both drag minimization and reconstruction design clearly show that the use of adaptive mesh refinement reduces the computational cost and also produces a better solution.