Toric varieties have become of major importance in algebraic geometry. A related but more general notion is that of a toric manifold, developed by M. W. Davis and T. Januszkiewicz [Duke Math. J. 62, No. 2, 417-451 (1991; Zbl 0733.52006)]. The authors have systematically studied the complex cobordism of such manifolds [Geom. Topol. 2, 79-101 (1998; Zbl 0907.57025)]. In this note they prove that every complex cobordism class contains a toric manifold equipped with a suitable complex structure. The proof uses the fact that connected sums of toric manifolds are toric, together with constructions of flag bundles from [the authors, loc. cit.]. Amusingly, the Milnor manifolds which are normally used in geometric constructions of generating sets are not toric. This result is interesting in part because it is still unknown if every complex cobordism class contains a connected projective variety. The greater flexibility obtained through using toric manifolds seems to make such results more accessible. There are a few unfortunate mistranslations of the Russian version: the repeated phrase ‘class of cobordisms’ should surely be ‘cobordism class’.