Simkins started playing jazz in his early teens. His first instrument was drums, but he quickly changed to the alto saxophone. He turned professional in 1977.
His early work included time with the Harry Strutters Hot Rhythm Orchestra and the Temperance Seven, but his principal stylistic influences have been the American alto player Lee Konitz and tenor player Warne Marsh.
Geoff is also a respected teacher, and as well as running improvisation courses in Brighton, UK, he has been a regular tutor at the famous Glamorgan Jazz Summer School in Wales (now discontinued, but later held at Trinity College of Music in London and, from 2012, at the Royal Welsh College of Music and Drama in Cardiff). In 1994 he was invited to be a tutor at the Czech Jazz Summer School and appeared at clubs in Prague and on Czech Television.