In 1967, the first radio pulsar was discovered by Jocelyn Bell and Anthony Hewish (for which Hewish received the Nobel Prize in 1974). Pulsars were quickly identified as neutron stars, the incredibly compressed remnants of the supernova explosion of stars.
In 1974, Russell Hulse and Joseph Taylor discovered the first binary pulsar PSR 1913+16: one of two neutron stars orbiting each other every 8 hours. The general theory of relativity predicts that as the stars orbit each other, they stir spacetime around them and radiate gravitational waves, causing them to spiral together.
In 1993, Taylor and Hulse received the Nobel prize for showing that since 1974 the neutron stars have been spiraling towards each other at exactly the rate Einstein's theory predicts, due to the emission of gravitational waves. LIGO and LISA are designed to directly detect the gravitational waves from binary systems like this.
From Beyond Einstein roadmap