Abstract: I will report on joint work with Changliang Wang on the linear stability question for Einstein metrics with positive scalar curvature, including families which admit a non-trivial Killing spinor. Previously, it had been shown by X. Dai, X. Wang, and G. Wei that compact Kahler-Einstein manifolds with non-positive scalar curvature and compact manifolds admitting a non-trivial parallel spinor are linearly stable. As well, Changliang Wang recently showed that complete manifolds admitting a non-trivial imaginary Killing spinor are also linearly stable.