  We show that nearly extremal Kerr black holes have two distinct sets of
quasinormal modes, which we call zero-damping modes (ZDMs) and damped modes
(DMs). The ZDMs exist for all harmonic indices $l$ and $m \ge 0$, and their
frequencies cluster onto the real axis in the extremal limit. The DMs have
nonzero damping for all black hole spins; they exist for all counterrotating
modes ($m<0$) and for corotating modes with $0\leq \mu\lesssim \mu_c=0.74$ (in
the eikonal limit), where $\mu\equiv m/(l+1/2)$. When the two families coexist,
ZDMs and DMs merge to form a single set of quasinormal modes as the black hole
spin decreases. Using the effective potential for perturbations of the Kerr
spacetime, we give intuitive explanations for the absence of DMs in certain
areas of the spectrum and for the branching of the spectrum into ZDMs and DMs
at large spins.
