Planxty are: 
 

Christy Moore	http://celtic.stanford.edu/artists/Christy_Moore.html
Donal Lunny	http://celtic.stanford.edu/artists/Donal_Lunny.html
Andy Irvine	http://celtic.stanford.edu/artists/Andy_Irvine.html
In 1978: Matt Molloy	http://celtic.stanford.edu/artists/Matt_Molly.html
Liam O’Flynn	Originally from the Bothy Band

(then Paul Brady in 1974)

Sometimes:  Tony Linnane, Bill Whelan (now of 
Riverdance fame)

For a taster, this is a quote from another site whose author knows more 
about this wonderful Irish folk group (now 25 years gone!) than I do:

"Planxty came about in 1972 when folk singer Christy Moore gathered a 
bunch of musicians for his upcoming album Prosperous. From this 
grouping, Moore, along with traditional uilleann piper LiamO'Flynn, 
Dónal Lunny and Andy Irvine, formed Planxty (the name refers to a tune 
composed for a patron,particularly associated with O'Carolan). Their 
first eponymous album, known generally as the "BlackAlbum" for it's 
cover art, stands as a landmark in Irish folk music, mixing authentic 
traditional music, withthe fire of modern folk, and the ballad singing that 
the likes of the Dubliners and the Clancys werepopularising. The 
harmonies and counterpoint of the group were relatively new to Irish 
music, whichtraditionally had been based on solo melodies. The singing 
of Moore and Irvine was backed by O'Flynn onuileann pipes and 
whistles, as well as guitar, bouzouki, mandolin and bodhran."

Check out http://celtic.stanford.edu/artists/Planxty.html
Discography:   See also 
http://celtic.stanford.edu/artists/Christy_Moore/pdiscog.html

Albums
 

1972   Planxty	Polydor 2383186     Shanachie 79009
1973   The well below the valley	Polydor 2383232     Shanachie 79010
1974   Cold blow and the rainy night	Polydor 2383301     Shanachie 79011
1976   The Planxty collection	Polydor 2389397     Shanachie 79012
1979   After the break	Tara 3001
1980   The woman I loved so well	Tara 3005
1982   Words & Music	WEA Ireland 2401011 Shanachie 79035
Singles
1972   Three drunken maidens	EMI
1972   The cliffs of Doneen	Polydor
1981?  Timedance/Nancy spain [12"]	WEA IR28207