The objective of this paper is to investigate the impact of two commonly used turbulence models with three different shape parameterization techniques in Reynolds-averaged Navier-Stokes simulations on two-dimensional optimum design obtained with gradient-based deterministic and robust aerodynamic shape optimization in transonic, viscous, turbulent flow. The main contribution of this study to aerodynamic design area is to evaluate the impact of turbulence models and shape parameterization techniques on the computational cost, optimal shape, and its performance obtained with robust optimization under uncertainty. The two turbulence models investigated include Spalart-Allmaras and Menter's shear-stress transport models. Hicks-Henne bump functions, B-spline curves, and free-form deformation are used as shape parameterization techniques. The results of the current study show that the shape parameterization technique has a 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 robustness of the final design obtained with stochastic optimization approach is also demonstrated over the Mach number range considered as the uncertain operating condition.