Born in 1947, Molloy is a native of county Roscommon, in the western area of Ireland. Molloy is widely recognized as the best flute player in Irish traditional music and was formerly in The Bothy Band and Planxty.

He started playing at the age of 12, and participated in traditional music festivals (called fleadhanna cheoil) where he won numerous competitions. In 1964, he moved to Dublin where he not only performed in clubs but also studied aeronautical engineering. He met Paddy Moloney at the Old Sheiling and soon the two were playing duets around town.

By the 1970s, Molloy was working as an airline mechanic and playing music on the side. When the traditional music scene boomed, he tried his hand at being a professional musician after securing a six-month leave of absence from his company. He never returned. He was a founding member of The Bothy Band, through which he became known as Irish music's premier flute player. Shortly after the group disbanded in 1979, Molloy joined Planxty, before becoming one of The Chieftains.

Discography:

The First Album: 1975, with The Bothy Band (Mulligan Lun 002/Green Linnet 3011, 1975)
Old Hag You Have Killed Me, with The Bothy Band (Mulligan/ Green Linnet 3005, 1976)
Out of the Wind, Into the Sun, with The Bothy Band (Polydor 2383456/Green Linnet 3013, 1977)
After Hours-Live in Paris, with The Bothy Band (Green Linnet 3016, 1979)
After the Break, with Planxty (1979)

With The Chieftains:

The Chieftains 9: Boil the Breakfast Early (1979)
The Chieftains 10: Cotton-Eyed Joe (1980)
The Year of the French (1981)
The Grey Fox (1982), soundtrack to The Grey Fox
The Chieftains in China (1985)
Ballad of the Irish Horse (1986)
Celtic Wedding (1987)
In Ireland, with James Galway (1987)
Irish Heartbeat, with Van Morrison (1988)
The Tailor Of Gloucester (1988)
A Chieftains Celebration (1989)
Over the Sea To Skye: The Celtic Connection (1990)
The Bells of Dublin (1991)
Another Country (1992)
An Irish Evening (1992)
The Celtic Harp: A Tribute To Edward Bunting (1993)
The Long Black Veil (1995)
Film Cuts (1996)
Santiago (1996)
Long Journey Home (1998)
Fire in the Kitchen (1998)
Silent Night: A Christmas in Rome (1998)
Tears of Stone (1999)
Water From the Well (2000)
The Wide World Over (2002)
Down the Old Plank Road: The Nashville Sessions (2002)
Further Down the Old Plank Road (2003)
Live From Dublin: A Tribute To Derek Bell (2005)
The Essential Chieftains (2006)
San Patricio (2010)