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**Abstract.** Informally, structural properties are usually characterized in one of two ways: either as properties expressible purely in terms of the primitive relations of mathematical theories, or as the properties that hold of all structurally similar mathematical objects. In this paper we present two formal explications of structural properties, corresponding to these two informal characterizations. Based on this, we discuss the relation between the two explications. As will be shown, the two characterizations do not determine the same class of mathematical properties. From this observation we draw some philosophical conclusions about the possibility of a “correct” analysis of structural properties.