Regular semigroups with multiplicative inverse transversals were first studied by Blyth and McFadden in 1982.In this paper, we extend the concept of inverse transversals to some kind of quasi-ideal adequate transversals of abundant semigroups.Some characterization theorems of abundant semigroups with different kinds of quasiideal adequate transversals are obtained. In particular, the structure theory of abundant semigroups having different kinds of quasi-ideal adequate transversals is established. Some results obtained by Blyth, McAlister and McFadden on inverse transversals in regular semigroups are hence extended to abundant semigroups.