  This note is concerned with the geometric classification of connected Lie
groups of dimension three or less, endowed with left-invariant Riemannian
metrics. On the one hand, assembling results from the literature, we give a
review of the complete classification of such groups up to quasi-isometries and
we compare the quasi-isometric classification with the bi-Lipschitz
classification. On the other hand, we study the problem whether two
quasi-isometrically equivalent Lie groups may be made isometric if equipped
with suitable left-invariant Riemannian metrics. We show that this is the case
for three-dimensional simply connected groups, but it is not true in general
for multiply connected groups. The counterexample also demonstrates that `may
be made isometric' is not a transitive relation.
