The biggest and most active hunt for life outside Earth started in 1960, when Frank Drake pointed the Green Bank radio telescope in West Virginia towards the star Tau Ceti. He was looking for anomalous radio signals that could have been sent by intelligent life. Eventually, his idea turned into SETI (standing for Search for Extra Terrestrial Intelligence), which used the downtime on radar telescopes around the world to scour the sky for any signals.
In 1974, astronomers intentionally transmitted a 210-byte message from the Arecibo Observatory in Puerto Rico in the hopes of signaling a civilization in the globular star cluster M13. The message contained fundamental information about humans and our corner of the universe, such as the atomic numbers of key elements and the chemical structure of DNA. But this sort of active communication has been rare. Astronomers mostly rely on passive communication -- listening for transmissions sent by alien civilizations.