The dynamics of solutions to the Einstein equations is richer in dimensions higher than $3+1$. In contrast with the classical stability of stationary, asymptotically flat black hole solutions in $3+1$ dimensions, some families of higher dimensional black holes suffer from dynamical instabilities. I will discuss a subtle instability affecting a wide class of higher dimensional black holes which has not been previously observed in the literature. This new instability is, in a sense, more fundamental than the other known instability phenomena in higher dimensions and can be related to a precise geometric property of the class of spacetimes considered.