The objective of this paper is to investigate the impact of two commonly used turbulence models in Reynolds Averaged Navier-Stokes simulations on the optimum design obtained with the gradient-based deterministic and robust aerodynamic shape optimization in transonic, viscous, turbulent flow. The robust design is performed under the variation of Mach number defined as the uncertain variable. The impact of each turbulence model is evaluated in terms of the computational cost and the difference in the shape and the performance of final design obtained with different turbulence models. The two turbulence models investigated include Spalart-Allmaras and the Menter's Shear Stress Transport models. In this study, Hicks-Henne bump functions, B-Spline curves and Free-Form Deformation are utilized as shape parameterization techniques. The results of the current study show that the shape parameterization technique has larger impact on the computational cost compared to the turbulence model in both deterministic and robust design. Robust design tends to reduce the impact of the turbulence model selection on the optimum shape and performance, whereas the turbulence model becomes important for the deterministic design at off-design conditions. In this study, the improvement of the robustness of the final design obtained with stochastic optimization approach is also demonstrated over the Mach number range considered as the uncertain operating condition.