Uncertainty quantification (UQ) is gaining in maturity and importance in engineering analysis. While historical engineering analysis and design methods have relied heavily on safety factors (SF) with built-in conservatism, modern approaches require detailed assessment of reliability to provide optimized and balanced designs. This paper presents methodologies that support the transition toward this type of approach. Fundamental concepts are described for UQ in general engineering analysis. These include consideration of the sources of uncertainty and their categorization. Of particular importance are the categorization of aleatory and epistemic uncertainties and their separate propagation through an UQ analysis. This familiar concept is referred to here as a “two-dimensional” approach, and it provides for the assessment of both the probability of a predicted occurrence and the credibility in that prediction. Unique to the approach presented here is the adaptation of the concept of a bounding probability box to that of a credible probability box. This requires estimates for probability distributions related to all uncertainties both aleatory and epistemic. The propagation of these distributions through the uncertainty analysis provides for the assessment of probability related to the system response, along with a quantification of credibility in that prediction. Details of a generalized methodology for UQ in this framework are presented, and approaches for interpreting results are described. Illustrative examples are presented.