  We show that rotating black holes do not experience any tidal deformation
when they are perturbed by a weak and adiabatic gravitational field. The tidal
deformability of an object is quantified by the so-called "Love numbers", which
describe the object's linear response to its external tidal field. In this
work, we compute the Love numbers of Kerr black holes and find that they vanish
identically. We also compute the dissipative part of the black hole's tidal
response, which is non-vanishing due to the absorptive nature of the event
horizon. Our results hold for arbitrary values of black hole spin, for both the
electric-type and magnetic-type perturbations, and to all orders in the
multipole expansion of the tidal field. The boundary conditions at the event
horizon and at asymptotic infinity are incorporated in our study, as they are
crucial for understanding the way in which these tidal effects are mapped onto
gravitational-wave observables. In closing, we address the ambiguity issue of
Love numbers in General Relativity, which we argue is resolved when those
boundary conditions are taken into account. Our findings provide essential
inputs for current efforts to probe the nature of compact objects through the
gravitational waves emitted by binary systems.
