It is an important step in deign under uncertainty to select an appropriate uncertainty propagation (UP) method considering the characteristics of the engineering systems at hand, the required level of UP associated with the probabilistic design scenario, and the required accuracy and efficiency levels. Many uncertainty propagation methods have been developed in various fields, however, there is a lack of good understanding of their relative merits. In this paper, a comparative study on the performances of several UP methods, including a few recent methods that have received growing attention, is performed. The full factorial numerical integration (FFNI), the univariate dimension reduction method (UDR), and the polynomial chaos expansion (PCE) are implemented and applied to several test problems with different settings of the performance nonlinearity, distribution types of input random variables, and the magnitude of input uncertainty. The performances of those methods are compared in moment estimation, tail probability calculation, and the probability density function (PDF) construction. It is found that the FFNI with the moment matching quadrature rule shows good accuracy but the computational cost becomes prohibitive as the number of input random variables increases. The accuracy and efficiency of the UDR method for moment estimations appear to be superior when there is no significant interaction effect in the performance function. Both FFNI and UDR are very robust against the non-normality of input variables. The PCE is implemented in combination with FFNI for coefficients estimation. The PCE method is shown to be a useful approach when a complete PDF description is desired. Inverse Rosenblatt transformation is used to treat non-normal inputs of PCE, however, it is shown that the transformation may result in the degradation of accuracy of PCE. It is also shown that in black-box type of system the performance and convergence of PCE highly depend on the method adopted to estimate its coefficients.