In 1966 Frank Hennessy and Dave Burns, both from Cardiff's Irish community, won the "Spotlight on Youth" competition organised by Cardiff City Council. They subsequently added Paul Powell (on banjo, vocals) to the line up and turned professional. In the late 60's the group went to Ireland, where, according to Frank Hennessy, "that pressure came…to sing a song or two in Welsh, to prove that they were different from the countless other folk and ballad groups which flourished at that time.". However The Hennessys didn't speak Welsh and the group "enlisted the help of one of Wales' finest folk musicologists, Meredydd Evans, and so began a long and fruitful friendship which saw the group including much more Welsh material in their repertoire." Additionally, Rhydderch Jones translated many of the group's favourite songs into Welsh and "then spent hours and hours trying to teach the boys to sing them properly."
In 1972, founder member Powell left the group and for a time Frank and Dave continued on as a duo. Aloma Jones (of Tony ac Aloma joined the group for a period in the early 70's. Tom Edwards (double bass) joined in 1979, and continued with the group until 1991, when he had to retire due to ill health. The current line up of the Hennessys is Frank (guitar, main vocals and composer), Dave (mandolin and vocals), and Iolo Jones (fiddle).
In addition, Dave Burns and Iolo Jones were founding members of the Welsh band, Ar Log. Both left the band the band in the early 1980's, but have since returned and continue to play with both The Hennessys and Ar Log.