This program describes UMLFA, a FORTRAN IV computer program for performing unrestricted maximum likelihood factor analysis. The program performs a factor analysis of a given correlation matrix. The factor loadings and the unique variances are estimated by Lawley's method of maximum likelihood. The computational procedure has been developed by the author and makes use of the method of Fletcher and Powell for numerical minimalization of a function. Any number of factors can be extracted and each factor matrix is rotated using Kaiser's varimax method. The goodness of fit of the maximum likelihood solution is tested by Lawley's chi- square test based on the likelihood ratio technique. Section 1 describes the basic assumptions underlying the maximum likelihood method and the chi-square test of goodness of fit, the distinction between proper and improper maximum likelihood solutions, and the basic features of the computational procedure. Sections 2 and 3 describe the input data and the output results--the standard output, the correlations, technical output, intermediate results, and punched factor matrix--and present a flow chart of the program. Appendices present an example of input data and the FORTRAN list. Revised Edition.