Gravitational waves promise to give answers to a number of fundamental
questions like the nature of gravity in the regime of strong
gravitational fields, the Kerr hypothesis, the dark matter and dark
energy puzzle, etc. The next generation of gravitational wave
detectors will observe hundreds of thousands of events per year
starting the era of high-precision gravitational wave astrophysics. In
the present talk, I will discuss what are our present and future
capabilities to test fundamental physics through gravitational waves.
I will focus primarily on merging compact binaries, as well as extreme
mass ratio inspirals and binary pulsars. Especially interesting are
the combinations of theories and astrophysical scenarios that give not
only small cumulative deviations with respect to general relativity
but also clear qualitatively different gravitational wave signatures
that are smoking guns for beyond-GR physics. Despite the rapid
development of the field, the theoretical modeling of such dynamical
processes beyond pure general relativity is still very limited leaving
space for exciting new discoveries.