The bus hit the woman at about 10:30 a.m. as it was going south on Webster Street toward University Avenue, according to police.
She was pronounced dead at the scene. Her name is being withheld pending notification of her family, police said.
The bus driver was taken to Kaiser Permanente in Redwood City to be treated for hypertension, police said. Two passengers were on the bus at the time.
Police said University Avenue will be closed in both directions between Middlefield Road and Cowper Street until about 3:30 p.m. while Palo Alto police and the San Mateo County sheriff's Transit Police Bureau investigate.