The dynamics of precessing black-hole binaries in the post-Newtonian regime is deeply characterized by a timescale hierarchy: the orbital timescale is very short compared to the spin-precession timescale which, in turn, is much shorter than the radiation-reaction timescale on which the orbit is shrinking due to gravitational-wave emission. We “average the average” and exploit both these timescale separations to greatly simplify the binary phenomenology. This procedure uncovers some nice surprises: morphological transitions, discontinuous limits, new resonant phenomena, and the dynamical instability of binaries with (anti)aligned spins. Double averaging also lead to impressive computational speed-up: post-Newtonian inspirals can now be computed from arbitrarily large separations, thus bridging the gap between astrophysics and numerical relativity. Already with the first detections, black hole spins are playing a key role to place astrophysical constraints on the observed populations, thus turning the promise of gravitational-wave astronomy into reality.