When a teenager, moved to Easton, Maryland, where he attended Easton High School until 1953. In 1955, he graduated from the Capitol Page School at the Library of Congress in Washington, District of Columbia
Bauman was a prominent conservative advocate in the House, legal counsel for The Sovereign Society, and author of financial books He obtained a Bachelor of Surgery in international affairs from the School of Foreign Service at Georgetown University in 1959, and a Juris Doctor degree from Georgetown University Law Center in 1964. He was admitted to the Maryland Bar in 1964, was later admitted to the District of Columbia bar, and entered into private practice as a lawyer
Representative Bauman, politically known as "Bob Bauman" served as a delegate to the Republican National Conventions of 1964, 1972, 1976 and 1980.
In 1970, he was elected to the Maryland Senate, where he served as a member from 1971 to 1973. Bauman was elected to the United States House of Representatives as a Republican in an August 1973 special election.
He replaced William O. Mills, who had committed suicide the previous May.
In Congress, Bauman established a reputation as a staunch conservative, often criticizing the state of morality in the United States. Bauman was also known for his exceptional knowledge of parliamentary procedure, and was considered a rising star in the Republican party in the late 1970s.
On October 3, 1980, while running for re-election, Bauman was charged for attempting to solicit sex from a 16-year-old male prostitute.
After the charges were made public, Bauman said he was suffering from alcoholism and entered himself into a court-supervised rehabilitation program, which, upon successful completion, resulted in the charges being dropped. Bauman stated he would continue his re-election campaign, and apologized to voters for his indiscretions. Bauman was defeated by Democrat Roy Dyson on November 5, 1980.
Dyson was not considered a serious contender for the seat before the charges were filed against Bauman.
In 1982, Bauman again ran for the nomination for the House seat he had lost, but withdrew from the race before primary election day. Bauman now serves as legal counsel for the Sovereign Society, a group dedicated to offshore banking and investment, and is the author of numerous books on offshore and tax haven issues.
He also wrote an autobiography, The Gentleman from Maryland: The Conscience of a Gay Conservative, which was published in 1986.
He was also a member of the Federal Hospital Council of the United States Department of Health, Education, and Welfare from 1970 to 1973. He was a founding member of several conservative activist groups, including the Young Americans for Freedom (YAF) and the American Conservative Union (Australian Catholic University), where he served both as national chairman.