In all, their record reads: played 16, won 7, lost 4, drawn 5.
They've progressed to the next round on three occasions, been eliminated three times, with the other double headers coming in the Champions League group stage.
Here BBC Sport looks at some of the memorable ties:
1966 - EUROPEAN CUP WINNERS' CUP SEMI-FINAL
14 April 1966 - Celtic 1-0 Liverpool 19 April 1966 - Liverpool 2-0 Celtic
Rangers, Kilmarnock and Dunfermline had already met English opposition in Europe.
Lennox's goal was not enough in 1966
But it was a double 'first' for Celtic as they were also aiming to reach their maiden European final.
Bobby Lennox's 52nd-minute goal gave Celtic a slender 1-0 advantage to take to Anfield.
But Bill Shankly's men went on to the final thanks to two goals in five second-half minutes from Tommy Smith and Geoff Strong.
Lennox thought he had scored a winning away goal in the last minute, only for the referee to disallow the effort for offside.
The Glasgow club made up for the disappointment the following year by winning the European Cup against Inter Milan.
1970 - EUROPEAN CUP SEMI-FINAL
01 April 1970 - Leeds United 0-1 Celtic 15 April 1970 - Celtic 2-1 Leeds United
The first match to carry the 'Battle of Britain' tag - between the champions of Scotland and England.
Celtic were aiming to reach their second European Cup final with Leeds United the team standing in their way.
Celtic beat the much-fancied Leeds on their own patch
Leeds were the team Celtic wanted to avoid as the overwhelming favourites to progress.
Don Revie's side had cost six times more to assemble than Jock Stein's side of local Scots.
But within a minute of kick-off at Elland Road, George Connelly struck to give Celtic a 1-0 away win.
And the Scottish champions finished the job in front of 136,000 fans at Hampden Park with John Hughes and Bobby Murdoch netting.
However, Stein was unable to claim his second European Cup as his team were beaten by Feyenoord in the final.
1997 - UEFA CUP FIRST ROUND
16 September 1997 - Celtic 2-2 Liverpool 30 September 1997 - Liverpool 0-0 Celtic
Celtic got off the worst possible start in the home leg, going behind with only five minutes on the clock.
McManaman scored a wonder-goal in Glasgow
A young Michael Owen, making only his eighth first-team appearance, put the visitors in front with a wonderfully cool chipped finish over home keeper Jonathan Gould.
Celtic came back level through a wonderful Jackie McNamara volley after the break, before Simon Donnelly scored from the spot with 15 minutes remaining.
But just as it looked like Liverpool were beaten, Steve McManaman took the ball on a run which started in his own half and ended in a sublime solo goal in the last minute.
That strike was to prove decisive as the return match on Merseyside ended 0-0, after Donnelly over-hit a lob with the goal at his mercy.
2003 - UEFA CUP QUARTER-FINAL
13 March 2003 - Celtic 1-1 Liverpool 20 March 2003 - Liverpool 0-2 Celtic
Earlier in the campaign Celtic had disposed of Blackburn Rovers 3-0 on aggregate, in what will be remembered for Rovers boss Graeme Souness remarking that the first leg at Celtic Park was like watching men against boys'.
Hartson was one of Celtic's heroes at Anfield
Liverpool provided a tougher test for Martin O'Neill's team, and although Henrik Larsson had given Celtic an early lead in Glasgow, Emile Heskey appeared to have struck a hammer blow with his equaliser.
With few giving Celtic much chance of leaving Anfield with a positive result, O'Neill told his players to show England and Europe that they deserved respect.
Alan Thompson's free-kick just before half-time, and an unforgettable John Hartson strike sealed the Glasgow club's passage to the last four, and they went on to lose in the final to Porto.
2006 - CHAMPIONS LEAGUE GROUP STAGE
13 September 2006 - Manchester United 3-2 Celtic 21 November 2006 - Celtic 1-0 Manchester United
This was Celtic's first meeting with English opposition in the Champions League era.
Nakamura's free-kick sent Celtic into the last 16
The Scots took a shock lead at Old Trafford through Jan Vennegoor of Hesselink, who controlled a long ball before shooting low past Edwin Van Der Sar.
A Louis Saha double put Sir Alex Ferguson's side in front, before Shunsuke Nakamura scored a direct free-kick to level at the break.
Celtic were undone minutes after the restart when Thomas Gravesen conceded possession cheaply, and the ball eventually fell to Ole Gunnar Solskjaer to seal a United win.
The return fixture was to prove historic for Gordon Strachan's Celtic. After riding out some heavy United pressure, Nakamura netted another audacious free-kick to send the Glasgow club to the last 16 for the first time.
Goalkeeper Artur Boruc's contribution was just as important as the Japanese's, as the Pole saved Saha's last-minute penalty.
CELTIC'S RECORD AGAINST ENGLISH TEAMS IN EUROPE
14 Apr 1966 - Celtic 1 (Lennox), Liverpool 0. 19 Apr 1966 - Liverpool 2 (Smith, Strong), Celtic 0.
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