Analytical and numerical solutions of the relativistic two-body problem have been essential to detecting and interpreting the several hundred gravitational-wave signals observed so far from binaries of black holes and neutron stars. In this talk, I will review key astrophysical, cosmological, and fundamental physics results from LIGO-Virgo-KAGRA observations, with an emphasis on the theoretical foundations that underpin them. I will then look ahead. Upcoming LIGO-Virgo-KAGRA runs and next-generation observatories (the Einstein Telescope and Cosmic Explorer on the ground, and the Laser Interferometer Space Antenna in space) will probe the universe back to cosmic dawn, test gravity in its most extreme regimes, and map the interior of neutron stars with unprecedented precision. Realizing this promise will require waveform models up to two orders of magnitude more accurate, along with major gains in computational efficiency. I will discuss the theoretical challenges we must overcome to interpret these observations reliably and avoid systematic biases that could lead to incorrect scientific conclusions.