  Black holes are the most compact objects in the Universe. According to
general relativity, black holes have a horizon that hides a singularity where
Einstein's theory breaks down. Recently, gravitational waves opened the
possibility to probe the existence of horizons and investigate the nature of
compact objects. This is of particular interest given some quantum-gravity
models which predict the presence of horizonless and singularity-free compact
objects. Such exotic compact objects can emit a different gravitational-wave
signal relative to the black hole case. In this thesis, we analyze the
stability of horizonless compact objects, and derive a generic framework to
compute their characteristic oscillation frequencies. We provide an analytical,
physically-motivated template to search for the gravitational-wave echoes
emitted by these objects in the late-time postmerger signal. Finally, we infer
how extreme mass-ratio inspirals observable by future gravitational-wave
detectors will allow for model-independent tests of the black hole paradigm.
