We describe mathematical and statistical models for factor invariance. We demonstrate that factor invariance is a condition of measurement invariance. In any study of change (as over age) measurement invariance is necessary for valid inference and interpretation. Two important forms of factorial invariance are distinguished: "configural" and "metric". Tests for factorial invariance and the range of tests from strong to weak are illustrated with multiple group factor and structural equation modeling analyses (with programs such as LISREL, COSAN, and RAM). The tests are for models of the organization and age changes of intellectual abilities. The models are derived from current theory of fluid (Gf) and crystallized (Gc) abilities. The models are made manifest with measurements of the WAIS-R in the standardization sample. Although this is a methodological paper, the key issues and major principles and conclusions are presented in basic English, devoid of technical details and obscure notation. Conceptual principles of multivariate methods of data analysis are presented in terms of substantive issues of importance for the science of the psychology of aging.