It is important to determine the existence and composition of homogeneous azeotropes in the analysis of phase behavior and in the synthesis and design of separation systems, from both theoretical and practical standpoints. A new method for reliably locating any and all homogeneous azeotropes for multicomponent mixtures is presented. The method also verifies the nonexistence of homogeneous azeotropes if none are present. The method is based on interval analysis, in particular an interval-Newton/generalized-bisection-algorithm providing a mathematical and computational guarantee that all azeotropes are located. This general-purpose technique can be applied in connection with any thermodynamic models. The technique is illustrated in several example problems using the Wilson, NRTL, and UNIQUAC activity-coefficient models.