There are so many traditions on Inauguration Day, but only one is required by law, and that is saying the Oath of Office. Taking the Oath with your hand on a book According to the U.S. Department of State,

Some presidents did not use a Bible to take the oath of office, including Theodore Roosevelt, who did not use anything when he was sworn into office in 1901, and John Quincy Adams, who chose a legal book for his 1825 swearing-in, to signify his responsibility to uphold the U.S. constitutional law.

The tradition of swearing the oath of office on a Bible stretches back to George Washington, but not all presidents have observed it.