This paper discusses simultaneous factor analysis in several populations, focusing on the study of similarities and differences in factor structures between different groups. A common scenario involves administering a battery of tests to samples from various populations. The paper presents a general model where parameters in factor analysis, such as factor loadings, variances, covariances, and unique variances, can be assigned arbitrary values or constrained to be equal. The model is estimated using maximum likelihood, yielding a large sample chi-square test of goodness of fit. By computing solutions under different specifications, various hypotheses can be tested. The method can handle any degree of invariance, from no invariance to full invariance. The number of tests and common factors need not be the same across groups, but a common core of tests is assumed. The paper also discusses identification of parameters, estimation and testing of the model, and a computer program for performing the analysis. It highlights the importance of scaling factors and observed variables, and provides a numerical illustration using data from Meredith's study. The results suggest that the factor structure may be invariant across populations, but further research is needed to determine the most appropriate model.This paper discusses simultaneous factor analysis in several populations, focusing on the study of similarities and differences in factor structures between different groups. A common scenario involves administering a battery of tests to samples from various populations. The paper presents a general model where parameters in factor analysis, such as factor loadings, variances, covariances, and unique variances, can be assigned arbitrary values or constrained to be equal. The model is estimated using maximum likelihood, yielding a large sample chi-square test of goodness of fit. By computing solutions under different specifications, various hypotheses can be tested. The method can handle any degree of invariance, from no invariance to full invariance. The number of tests and common factors need not be the same across groups, but a common core of tests is assumed. The paper also discusses identification of parameters, estimation and testing of the model, and a computer program for performing the analysis. It highlights the importance of scaling factors and observed variables, and provides a numerical illustration using data from Meredith's study. The results suggest that the factor structure may be invariant across populations, but further research is needed to determine the most appropriate model.