  We obtain a structure theorem for closed, cohomogeneity one Alexandrov spaces
and we classify closed, cohomogeneity one Alexandrov spaces in dimensions 3 and
4. As a corollary, we obtain the classification of closed, $n$-dimensional,
cohomogeneity one Alexandrov spaces admitting an isometric $T^{n-1}$ action. In
contrast to the 1- and 2-dimensional cases, where it is known that an
Alexandrov space is a topological manifold, in dimension 3 the classification
contains, in addition to the known cohomogeneity one manifolds, the spherical
suspension of $RP^2$, which is not a manifold.
