  We consider Lie groups equipped with arbitrary distances. We only assume that
the distance is left-invariant and induces the manifold topology. For brevity,
we call such object metric Lie groups. Apart from Riemannian Lie groups,
distinguished examples are sub-Riemannian Lie groups and, in particular, Carnot
groups equipped with Carnot-Carath\'eodory distances. We study the regularity
of isometries, i.e., distance-preserving homeomorphisms. Our first result is
the analyticity of such maps between metric Lie groups. The second result is
that if two metric Lie groups are connected and nilpotent then every isometry
between the groups is the composition of a left translation and an isomorphism.
There are counterexamples if one does not assume the groups to be either
connected or nilpotent. The first result is based on a solution of the Hilbert
5th problem by Montgomery and Zippin. The second result is proved, via the
first result, considering the Riemannian case, which for self-isometries was
solved by Wolf.
