SETI is an acronym that stands for "Search for Extra Terrestrial Intellegence. It is characterized mainly by use of radio telescopes to search for signals from other star systems.
The SETI Era began in 1959, when ideas were first developed for using microwave radio waves to communicate between the stars. Cornell University physicists Giuseppi Cocconi and Philip Morrison are credited for the first published article along these lines (In Nature Magazine).
The first microwave radio search for signals from other solar systems was performed in 1960 by Frank Drake. Drake searched two Sun like stars for signals using the 21 cm wavelength (1,420 MHz frequency). This is considered the "magic" frequency because it corresponds to the spectral line of neutral hydrogen, the most abundant element in the universe.
SETI (like most of space-related science) in the 1960's was dominated by the Soviet Union. It wasn't until the early 1970s that NASA got involved with Project Cyclops, a comprehensive study that provided an analysis of SETI science and technology issues. After this, the prospects of SETI became more clear (and reasonable) and more Americans started to do radio searches.
Some of these searches are very long running, and many still exist today. Some of the more famous (and long-running) searches are Planetary Society's Project META, the University of California's SERENDIP project, and a long-standing observing program at Ohio State University.
By the late-1970's two major programs developed very different strategies. One was the Targeted Search, detecting weak or sporadic signals. The other was the broad Sky Survey, scanning large portions of the sky periodically. In 1988, after a decade of study and preliminary design, NASA Headquarters formally adopted these strategy, and funded the programs.
Within a year, Congress terminated funding.
Since there is much to say on the subject, I have broken it down into several sections:
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