This paper is a comprehensive study of methods for assessing unidimensional expected utility functions. The paper describes the utility assessment process in decision analysis and then reviews problem formulation, sources of bias in preference judgments, and the analysis of risk attitudes. Two dozen utility assessment methods of which half appear for the first time are critically examined. These methods are grouped into preference comparison methods, probability equivalence methods, value equivalence methods, certainty equivalence methods, hybrid methods, paired-gamble methods, and other approaches. The paper emphasizes the nature of judgmental biases in comparing different assessment procedures. Since most multiattribute utility functions are decomposed into single-attribute functions, this study should facilitate such applications. The paper concludes with several directions for further developmental, empirical, and applied research.