Yarwood was born in Whyalla, South Australia, and has worked as a town planner. He became a councillor in the City of Adelaide in 2007, and in 2008 was elected to the position of Deputy Lord Mayor. He won the 2010 mayoral election with 3169 votes, 884 votes ahead of the second-placed candidate. At 39 years old, Yarwood is the youngest Lord Mayor in the city's history.
Stephen went to Norwood Morialta High School and graduated from the University of South Australia in 1993 with a Bachelor of Arts (Planning), he has also studied and been awarded a Masters of Business Administration.