  The tidal Love numbers (TLNs) encode the deformability of a self-gravitating
object immersed in a tidal environment and depend significantly both on the
object's internal structure and on the dynamics of the gravitational field. An
intriguing result in classical general relativity is the vanishing of the TLNs
of black holes. We extend this result in three ways, aiming at testing the
nature of compact objects: (i) we compute the TLNs of exotic compact objects,
including different families of boson stars, gravastars, wormholes, and other
toy models for quantum corrections at the horizon scale. In the black-hole
limit, we find a universal logarithmic dependence of the TLNs on the location
of the surface; (ii) we compute the TLNs of black holes beyond vacuum general
relativity, including Einstein-Maxwell, Brans-Dicke and Chern-Simons gravity;
(iii) We assess the ability of present and future gravitational-wave detectors
to measure the TLNs of these objects, including the first analysis of TLNs with
LISA. Both LIGO, ET and LISA can impose interesting constraints on boson stars,
while LISA is able to probe even extremely compact objects. We argue that the
TLNs provide a smoking gun of new physics at the horizon scale, and that future
gravitational-wave measurements of the TLNs in a binary inspiral provide a
novel way to test black holes and general relativity in the strong-field
regime.
