  We study homogeneous metric spaces, by which we mean connected, locally
compact metric spaces whose isometry group acts transitively. After a review of
some classical results, we use the Gleason-Iwasawa-Montgomery-Yamabe-Zippin
structure theory to show that for all positive $\epsilon$, each such space is
$(1,\epsilon)$-quasi-isometric to a connected metric Lie group. Next, we
develop the structure theory of Lie groups to show that every homogeneous
metric manifold is homeomorphically roughly isometric to a quotient space of a
connected amenable Lie group, and roughly isometric to a simply connected
solvable metric Lie group. Third, we investigate solvable metric Lie groups in
more detail, and expound on and extend work of Gordon and Wilson and of
Jablonski on these, showing, for instance, that connected, simply connected
solvable Lie groups may be made isometric if and only if they have the same
real-shadow. Finally, we extend a result of Kivioja and Le Donne to show that
homogeneous metric spaces that admit a metric dilation are all metric Lie
groups with an automorphic dilation.
