Crossley was educated at Queen Elizabeth Grammar School, Wakefield, and then went up to St John's College, Oxford.
He was a Harmsworth Senior Scholar at Merton College from 1960 to 1962, before taking up a one-year Junior Research Fellowship there; he received his DPhil and MA (Mathematics) in 1963.
[3] He was offered a Readership position and following a lecturing visit to Monash University in 1968, he was elected to a Chair in Pure Mathematics.
He accepted this position and as of 2010, Crossley continues to be active at Monash University where he serves through its Faculty of Information Technology.
Many of Crossley's doctoral students[5] have gone on to be professors themselves and have written books in the field of mathematics or computing, including Peter Aczel, Wilfrid Hodges, John Lane Bell and Rod Downey.