Einstein metrics are motivated by Einsteins field equations of
general relativity. Because of their many intriguing properties
they are also an important research topic in Riemannian
geometry.

Einstein metrics can be characterised as critical points of the
total scalar curvature functional. They are always
saddle points. However, there are Einstein metrics which are
local maxima of the functional restricted to metrics of fixed
volume and constant scalar curvature. These are by definition stable.
Stability can also be described by a spectral condition for a
Laplace type operator on symmetric 2-tensors. Moreover, stability
is related to Perelman's $\nu$ entropy and dynamical stability with
respect to the Ricci flow.

In my talk I will give an introduction to Einstein metrics and
discuss the stability condition in some detail. I also plan to
present a few recent results on the stability of symmetric spaces
and an interesting relation between instability and harmonic forms.
These results were obtained in joint work with G. Weingart resp.
with M. Wang and Ch. Wang.