  Superspinars are ultracompact objects whose mass M and angular momentum J
violate the Kerr bound (cJ/GM^2>1). Recent studies analyzed the observable
consequences of gravitational lensing and accretion around superspinars in
astrophysical scenarios. In this paper we investigate the dynamical stability
of superspinars to gravitational perturbations, considering either purely
reflecting or perfectly absorbing boundary conditions at the "surface" of the
superspinar. We find that these objects are unstable independently of the
boundary conditions, and that the instability is strongest for relatively small
values of the spin. Also, we give a physical interpretation of the various
instabilities that we find. Our results (together with the well-known fact that
accretion tends to spin superspinars down) imply that superspinars are very
unlikely astrophysical alternatives to black holes.
