We will report on recent progress towards the classification of
homogeneous Einstein metrics. While homogeneous Ricci-flat spaces are flat,
homogeneous Einstein spaces with negative Einstein constant are
diffeomorphic
to a Euclidean space, as predicted by the Alekseevskii conjecture. If
time permits we
will indicate some ideas of the proof. The classification of homogeneous
Einstein spaces
with positive Einstein constant is wide open, even though there are
general existence and
non-existence results.