Rockabilly never really went away.
The first revival occured in the late 60's, spearheaded by the band Sha Na Na. They were huge. They played Woodstock fer chrissakes. Being so close to the 50's, nobody could take it seriously.
In the UK, the rockabilly spirit was kept alive by the rockers. There were the mods, the rockers, and the teds, according to many punk rock history books.
The late 70's spawned The Cramps and The Blasters, who were underground giants, but achieved little pop success. They were lumped in with the punks, because they rocked.
The 80's brought The Stray Cats from the UK, who were really transplants from the US. They were lumped in with the new wavers, because they had poofy clothes and haircuts.
The 90's were dominated by The Reverend Horton Heat and Southern Culture on the Skids, and a pseudo-underground rockabilly scene started to grow into its own thing, thanks to MTV.
The most notable qualities about the rockabilly scene is that it's conservative, peculiarly white, and more concerned about cars and clothes than music.