This thesis deals with variable-complexity optimization, which consists of either using high- and low-fidelity models of the analysis or using variable parameterization of the optimization problem to reduce the computational cost of the optimization process. In this thesis, we present a variable-fidelity framework that is mathematically robust. We then present the results on analytical test cases for the framework and variable-parameterization method, which involves using different design variables during the course of the optimization process. The framework and the variable-parameterization method are then used to perform airfoil shape optimization. The variable-fidelity framework performed satisfactorily for most cases while a thorough mathematical study is needed for the variable-parameterization method.