Conway was a consistent member of the close knit Cat Stevens backing band throughout the mid-1970s. When Stevens gave up his pop music career at the close of the decade, Conway performed on Daydo; the short-lived solo album of singer-songwriterAlun Davies, another long-term member of Stevens' band.
Conway has secured the drummer's position in Fairport Convention (on a regular basis since 1998) and Pentangle (occasionally). In the years since 2006, Davies has re-joined with Stevens (now known as Yusuf Islam). Both Conway and Davies additionally play in a side project called "Good Men in the Jungle", which features Davies' daughter, Becky Moncurr.