  We show that the horofunction boundary of Teichm\"uller space with Thurston's
Lipschitz metric is the same as the Thurston boundary. We use this to determine
the isometry group of the Lipschitz metric, apart from in some exceptional
cases. We also show that the Teichm\"uller spaces of different surfaces, when
endowed with this metric, are not isometric, again with some possible
exceptions of low genus.
