Celtic have played Heart of Midlothian more times than any other club apart from the original Rangers. Unsurprisingly there is a storied Scottish Cup history betwixt the pair.
Head-to-Head
They’ve have met in 26 seasons and on 31 occasions in total, given replays. On average they meet every 5 seasons in the Cup.
Celtic lead the head to head with 19 wins to 7.

Celtic have only lost once in Edinburgh. Whilst Hearts, encouragingly for them, have won 6 times in Glasgow - 3 times at Celtic Park.
The matches generate a lot of goals. The ties average 3.3 goals. Three times there have been 7 goals in a match.
Hearts should be enthused by the fact that they have won 2 out of 3 Cup Finals versus the Hoops. The last final was in 1956. [bctt tweet="Indeed, Celtic have not beaten Hearts in a Scottish Cup Final for 112 years." username="Alan_Morrison67"]
Celtic have had the better of Semi Final’s with four wins out of five. However, Hearts won the latest edition in 2011/12.
Scores and Scorers
With thanks as always to the Celtic Wiki here are the scores and scorers (if you can fill in any gaps please inform the Celtic Wiki).
1900-39

1946 – 99

2000 – present

Kris Commons is the only player to score a hat trick in these ties, as part of a record equalling (magnitude of defeat for Hearts) 7-0 win in 2013/14.
Lisbon Lion Willie Wallace is the only player to have scored in three consecutive ties in the series – for Hearts! - in the early to mid-1960s.
Stevie Chalmers also scored in three different ties, but over four matches.
Many great names from both sides’ histories feature in the list of scorers.
Notable Matches
1900/01 – Celtic could have won 3 Scottish Cups in a row – a feat never achieved. Celtic lost to a late goal from Bell having fought back from 1-3. Only 16,000 at Ibrox attended due to the cost of tickets and poor weather.
1906/07 – Celtic’s first double. 50,000 at Hampden. Celtic forced 8 corners in 20 first half minutes as they took control. Celtic only scored thanks to a “doubtful” penalty through Orr. Celtic then scored 2 on the break.
1925/26 – a then record Tynecastle crowd of 51,000 saw Celtic win 4-0.
1932/33 - Semi Final – 87,000 at Hampden for 0-0 draw. The replay saw 63,756 and Celtic won an unimpressive 2-1 with reports noting “much energy was expended in a futile manner”. Not much changes.
1938/39 – Remarkably, to reach the Third Round Hearts had a 14-2 win versus Penicuik Athletic followed up by a 14-1 win over Elgin City. Strangely this is nowhere near their record win (21-0!). Celtic had scored 8 at Burntisland (8-3!) and 7 at Montrose (7-1). A Third Round replay was won in the last embers of extra time in front of 80,840 at Celtic Park. This was a record for a mid-week British match outside of Scottish Cup Final replays and Glasgow Derby New Year clashes. Hearts protested the winning goal from Divers claiming it had not gone over the line.
1955/56 – Hearts won the most recent final between the pair. A crowd of 132,842 witnessed a deserved victory over an injury hit Celtic side – the Edinburgh side’s first Cup win in 49 years. Apparently the Hearts ‘keeper never had a shot to save as Heart’s famed forward line of Crawford, Conn, Bauld and Wardhaugh triumphed in the Hampden wind.
1961/62 – A Third Round thriller as Celtic go behind, then ahead, are pegged back and win with a late 86th minute twice-taken penalty from Crerand. Hearts protested Chalmers goal as offside, claimed a Rodgers shot had crossed the line at 3-3 and bemoaned Crerand being allowed to retake a missed penalty due to encroachment. All from the infamous R H Davidson of Airdire!
1965/66 – Another exciting replayed tie. Celtic and Hearts shared 6 goals at a packed Tynecastle (over 45,000 in attendance). A Celtic side resembling the Lisbon Lions won easily with Wallace’s late consolation notable for being the third cup match running he had scored for Hearts against his future employers.
1971/72 – Yet another replayed tie. Heart’s disciplined display earned an 88th minute equaliser for Renton. Celtic had sat back and got what they deserved as Dalglish sat on the bench. The replay, played on a Monday as Celtic prepared for a European Cup Semi Final, was another tense affair. Macari’s header after 35 minutes settled it. The game was delayed 10 minutes near the end for 6 minutes due to “serious crowd disorder”.
1982/83 – The longest hiatus between Cup ties (11 years) ended in the 82/83 season. Hearts were not a Premier Division side in that season. The top division side won easily and the visitors were hindered by Willie Johnston receiving his 19th career red card. Nicholas scored 2 and assisted 1 as Celtic ran out easy winners.
1986/87 - The start of three cup ties in consecutive seasons, and Hearts only win against Celtic in the Scottish Cup at Tynecastle! Scoring legend John Robertson’s 80th minute winner ended a bruising game of few chances.
1987/88 – One of Celtic’s most famous victories. The Centenary Double hung in the balance as Hearts led the Semi Final going into the 88th minute. Goals from McGhee (88) and Walker in added time kept that magical season alive. Goalkeeper Smith dropped a corner allowing McGhee to stab home. Two minutes later, Smith again dropped a cross under challenge and Walker prodded home. Celtic’s unbeaten run that season was thus extended to 31 matches, where it ended – they lost their next league game – to Hearts!
1988/89 – Familiarity bred contempt as a third meeting in three seasons saw Celtic edge a Quarter Final. Three players were sent off (Tosh McKinlay and Alan McLaren for Hearts, Mick McCarthy for Celtic). Celtic were 2 up and had the whole second half 10 vs 9. Despite this Bannon scored a 72nd minute free kick as Celtic edged nervously home.
2000/01 – The first Cup encounter of the 21st Century saw Celtic take another step to what would be (in those days)! a rare Treble. Inevitably Larsson headed the winner, his 43rd of the season, from a Thompson cross late in the first half. Celtic had scored 18 goals against the Jambos that season but this was a battling display, but to no avail.
2003/04 – Once again Hearts ended a Cup match versus Celtic with 9 players as Maybury and Hartley saw Red. A goal down in 3 minutes, Hearts were never in the game. Petrov scored 2 and Larsson assisted both as well as bagging his customary goal late on. Future Celts Gordon, Pressley and Hartley all played that day.
2004/05 – Celtic won their 4th Semi Final against Hearts out of four. As O’Neill’s side limped towards Black Sunday, the consolation of another Cup Final, in the 120th edition of the tournament, was secured. A crowd of just over 38,000 saw Celtic lead through Sutton and Bellamy before Cesnauskis brought Hearts hope.
2011/12 – Hearts recorded their only Cup Semi Final win over the Celts. Celtic had wrapped up the league title with a 6-0 win against Kilmarnock the week before. An all Edinburgh final was secured with a late and controversial Beattie penalty. The ball was kicked onto Ledley’s arm from close range and the ex-Celt gleefully dispatched the penalty on 88 minutes. Also controversially, Hooper had headed an equaliser 2 minutes prior that looked offside.
2013/14 – A young Hearts side, suffering the real consequences of administration, were no match for a rampant Celtic. A record equalling quantum of defeat saw Commons become the only player to score a hat trick in these Scottish Cup encounters. A small crowed just over 10,000 also saw a rare Brown double.
2014/15 – A rejuvenated Hearts headed the First Division looking to bounce back to the Premier League. However, the gulf between the divisions was stark. Collum dismissed Gomis for a horror challenge after only 8 minutes and with Van Dijk scoring on 30, the game was up. Hearts were unlucky when Collum erroneously awarded a second half penalty after Guidetti slipped. Having to concentrate on the league, the Jambos were able to secure a quick return to the Premier.
Summary
So, now you know your history, you should expect:
•Goals! (3.3 per tie);
•Controversy! (many disputed goals given and not);
•Violence! (many “heated” encounters and Red Cards); and
•That Hearts generally perform well in Glasgow in the Scottish Cup against Celtic.