One of the most important considerations in psychological and educational assessment is the extent to which a test is free of bias and fair for groups with diverse backgrounds. Establishing measurement invariance (MI) of a test or items is a prerequisite for meaningful comparisons across groups as it ensures that test items do not function differently across groups. Demonstration of MI is particularly important in assessment settings where test scores are used in decision making. In this review, we begin with an overview of test bias and fairness, followed by a discussion of issues involving group classification, focusing on categorizations of race/ethnicity and sex/gender. We then describe procedures used to establish MI, detailing steps in the implementation of multigroup confirmatory factor analysis, and discussing recent developments in alternative procedures for establishing MI, such as the alignment method and moderated nonlinear factor analysis, which accommodate reconceptualization of group categorizations. Lastly, we discuss a variety of important statistical and conceptual issues to be considered in conducting multigroup confirmatory factor analysis and related methods and conclude with some recommendations for applications of these procedures. (PsycINFO Database Record (c) 2019 APA, all rights reserved).