This paper describes an algorithm for the computation of the homogeneous and heterogeneous azeotropes present in a multicomponent mixture. The approach (which is an extension of an approach developed by {Fidkowski}, {Malone} and {Doherty} (1993, Computers and {Chemical} {Engineering}, 17, 1141-1155) for the computation of homogeneous azeotropes) is independent of the representation of the nonideality of the system and the topology of the liquid-liquid region. Furthermore, the approach can be readily extended to handle any number of liquid phases in equilibrium. The second paper in this series discusses how the approach can be used to compute efficiently changes in equilibrium structure with parameters such as pressure (i.e., the appearance and disappearance of azeotropes and heteroazeotropes) and contains several numerical examples of the approach described in this paper.