  Astrophysical tests of general relativity belong to two categories: 1)
"internal", i.e. consistency tests within the theory (for example, tests that
astrophysical black holes are indeed described by the Kerr solution and its
perturbations), or 2) "external", i.e. tests of the many proposed extensions of
the theory. I review some ways in which astrophysical black holes can be used
as natural laboratories for both "internal" and "external" tests of general
relativity. The examples provided here (ringdown tests of the black hole
"no-hair" theorem, bosonic superradiant instabilities in rotating black holes
and gravitational-wave tests of massive scalar-tensor theories) are shamelessly
biased towards recent research by myself and my collaborators. Hopefully this
colloquial introduction aimed mainly at astrophysicists will convince skeptics
(if there are any) that space-based detectors will be crucial to study
fundamental physics through gravitational-wave observations.
