Bryan was born in Washington, D.C., and graduated from the University of Nevada at Reno in 1959 where he was a member of Alpha Tau Omega and the president of ASUN.

In 1982, Bryan challenged incumbent Republican Nevada Governor Robert List, who was running for reelection.

Bryan was an opponent of Search for extraterrestrial intelligence (SETI), stating: "As of today millions have been spent and we have yet to bag a single little green fellow.

He introduced an amendment to the 1994 budget that secured the cancellation of the High Resolution Microwave Survey and terminated NASA's SETI efforts less than one year after their launch.

Though the Yucca Mountain nuclear waste repository would be built during Bryan's time in the Senate, his opposition, delayed any actual storage from occurring.