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Spin precession occurs in binary black holes whose spins are misaligned withthe orbital angular momentum. Otherwise, the spin configuration is constant andthe subsequent binary dynamics and gravitational-wave emission are muchsimpler. We summarize a series of works which has shown that, while three ofthe aligned configurations are stable equilibria, the `up-down' configuration,in which the heavier (lighter) black hole is (anti) aligned with the orbitalangular momentum, is unstable when perturbed; at a critical point in theinspiral the black hole spins begin to tilt wildly as precession takes over. Wepresent two equivalent approaches to derive the instability onset based onmultitimescale post-Newtonian techniques, and point out that the instabilityhas a predictable endpoint. Finally, we demonstrate the presence of thisprecessional instability in the strong-field regime of numerical relativitywith simulations of aligned-spin binaries lasting $\sim100$ orbits beforemerger. The spins of up-down systems can tilt by $\sim90^\circ$, leaving anotable imprint in the emitted gravitational-wave signals and providing apossible mechanism to form precessing systems in astrophysical environmentsfrom which sources are preferentially born with (anti) aligned spins.

Unstable binary black-hole spins: post-Newtonian theory and numerical relativity