Though the film's events, loosely based on the ABSCAM sting operation, takes place during 1978-1980, most of the music is from earlier on in the decade. And some of the tunes date back much, much further. Jacobs comments on the process of song choices in this interview.
The official soundtrack album (above) features half of the songs heard in the film and it makes for a pretty good listen. But HERC wanted more (Steely Dan, Chicago, Todd Rundgren, Temptations - to name a few) and thanks to Spotify, he got it:
Spotify user Ludwig Mattsson put together what he calls American Hustle - Complete Soundtrack and it certainly lives up to that title with 28 tracks.