  This paper is concerned with embeddings of homogeneous spaces into Euclidean
spaces. We show that any homogeneous metric space can be embedded into a
Hilbert space using an almost bi-Lipschitz mapping (bi-Lipschitz to within
logarithmic corrections). The image of this set is no longer homogeneous, but
`almost homogeneous'. We therefore study the problem of embedding an almost
homogeneous subset $X$ of a Hilbert space $H$ into a finite-dimensional
Euclidean space. In fact we show that if $X$ is a compact subset of a Banach
space and $X-X$ is almost homogeneous then, for $N$ sufficiently large, a
prevalent set of linear maps from $X$ into $\Re^N$ are almost bi-Lipschitz
between $X$ and its image. We are then able to use the Kuratowski embedding of
$(X,d)$ into $L^\infty(X)$ to prove a similar result for compact metric spaces.
