Summary: The notion of a randomization of a first-order structure was introduced by H. J. Keisler in the paper [“Randomizing a model”, Adv. Math. 143, No. 1, 124–158 (1999; Zbl 0924.03065)]. The idea was to form a new structure whose elements are random elements of the original first-order structure. In this paper we treat randomizations as continuous structures in the sense of Ben Yaacov and Usvyatsov. In this setting, the earlier results show that the randomization of a complete first-order theory is a complete theory in continuous logic that admits elimination of quantifiers and has a natural set of axioms. We show that the randomization operation preserves the properties of being omega-categorical, omega-stable, and stable.