  This article is concerned with the stability of triharmonic maps and in
particular triharmonic hypersurfaces. After deriving a number of general
statements on the stability of triharmonic maps we focus on the stability of
triharmonic hypersurfaces in space forms, where we pay special attention to
their normal stability.
  We show that triharmonic hypersurfaces of constant mean curvature in
Euclidean space are weakly stable with respect to normal variations while
triharmonic hypersurfaces of constant mean curvature in hyperbolic space are
stable with respect to normal variations.
  For the case of a spherical target we show that the normal index of the small
proper triharmonic hypersphere is equal to one and make some comments on the
normal stability of the proper triharmonic Clifford torus.
