Cost-effectiveness analysis (CEA) is one of the main tools of economic evaluation. Every CEA is based on a number of assumptions, some of which may not be accurate, introducing uncertainty. Sensitivity analysis (SA) formalizes ways to measure and evaluate this uncertainty. Specific sources of uncertainty in CEA have been noted by various researchers. In this work, we consolidate across all sources of uncertainty, discuss the imbalanced attention to SA across different sources, and discuss criteria for conducting and reporting SA to help bridge the gap between guidelines and practice. Guidelines on how to perform SA have been published for many years in response to requests for greater standardization among researchers. Decision makers tasked with reviewing new health technologies also seem to appreciate the additional information conveyed by a robust SA, including the attention to important patient subgroups. Yet, past reviews have shown that there is a substantial gap between the guidelines' suggestions and the quality of SA in the field. Past reviews have also focused on one or two but not all three sources of uncertainty. The objective of our work is to comprehensively review all different sources of uncertainty and provide a concise set of criteria for conducting and presenting SA, stratified by common modelling approaches, including decision analysis and regression models. We first provide an overview of the three sources of uncertainty in a CEA (parameter, structural and methodological), including patient heterogeneity. We then present results from a literature review of the conduct and reporting of SA based on 406 CEA articles published between 2000 and mid-2009. We find that a minority of papers addressed at least two of the three sources of uncertainty, with no change over time. On the other hand, the use of some sophisticated techniques, such as probabilistic SA, has surged over the past 10 years. Lastly, we identify criteria for reporting uncertainty-robust SA and also discuss how to conduct SA and how to improve the reporting of SA for decision makers. We recommend that researchers take a more comprehensive view of uncertainty when planning SA for an economic evaluation.