  Levi subgroup actions on Schubert varieties are studied. In the case of
partial flag varieties, the horospherical actions are determined. This leads to
a characterization of the toroidal and horospherical partial flag varieties
with Picard number 1. In the more general case, we provide a set of necessary
conditions for the action of a Levi subgroup on a Schubert variety to be
toroidal. The singular locus of a (co)minuscule Schubert variety is shown to
contain all the $L_{max}$-stable Schubert subvarieties, where $L_{max}$ is the
standard Levi subgroup of the maximal parabolic which acts on the Schubert
variety by left multiplication. In type A, the effect of the Billey-Postnikov
decomposition on toroidal Schubert varieties is obtained.
