  We classify maximally supersymmetric backgrounds (vacua) of chiral (1,0) and
(2,0) supergravities in six dimensions and, by reduction, also those of the
minimal N=2 supergravity in five dimensions. Up to R-symmetry, the (2,0) vacua
are in one-to-one correspondence with (1,0) vacua, and these in turn are
locally isometric to Lie groups admitting a bi-invariant lorentzian metric with
anti-selfdual parallelising torsion, which we classify. We then show that the
five-dimensional vacua are homogeneous spaces arising canonically as the spaces
of right cosets of spacelike one-parameter subgroups.
