Who will forget those scenes from Italia 90, when David O'Leary's penalty against Romania sent the Republic of Ireland into the quarter-finals of the World Cup?
Or Ray Houghton's lobbed winner against Italy at the Giants Stadium, New York in 1994?
And what about in 2002, when Robbie Keane's late equaliser against Germany sent Mick McCarthy into that famous Irish jig on the touchline in Kashima?
Those days seemed long gone when the Republic failed dismally to reach the final stages of Euro 2008, and Steve Staunton paid for that failure with his job as manager of the national team.
As Ireland prepared for another attempt at qualification for a major tournament, the World Cup in South Africa in 2010, what they needed to guide them was a man of great experience.
Someone who has managed at the very highest level. Someone who would be respected by every player in the squad. Most of all, someone who take the team forward.
They got him. Giovanni Trapattoni was that man.
Houghton's super strike v Italy 1994
For Stoke City midfielder Glenn Whelan, the arrival of Trapattoni as the Republic of Ireland's national coach has given him an unexpected chance to establish himself as an international regular.
Last year was quite a period for the former Manchester City and Sheffield Wednesday player.
He helped the Potters back into the Premier League for the first time in 23 years, and made a long-awaited breakthrough into the Republic squad.
He told BBC Radio Stoke: "He's made a massive impression so far. He's got a lot of experience, he's managed some top clubs around the world with some top players.
"Obviously if you go and train with him, and learn just a little bit from him, it will be great for you as a player.
Keane's late equaliser v Germany 2002
The appointment of Trapattoni signalled a change in approach from the Football Association of Ireland, who had tended to go down the route of handing the job to coaches who were inexperienced at the top level.
It had not worked with Staunton, or his predecessor Brian Kerr.
In stark contrast, Trapattoni boasted one of the most glittering CVs in world football.
After a playing career that contained almost 300 club appearances as a tough-tackling centre-back for AC Milan and Varese and 17 caps for Italy, Trapattoni embarked on a managerial career that has spanned nearly four decades.
AC Milan (twice), Juventus (twice), Inter Milan, Bayern Munich (twice), Cagliari, Fiorentina, Benfica, Stuttgart and Red Bull Salzburg - all big clubs that have been managed by the unassuming 70-year-old.
So it comes as no surprise that Trapattoni is the most successful coach in the history of Italian football.
Glenn Whelan's only international goal came against Georgia in October
Whelan, 25, who made his debut in Trapattoni's first match in charge, the 1-1 draw against Serbia at Croke Park, has played a part in all eight of the games that the Italian has been at the helm.
"Trapattoni has seen something in me and hopefully I've been putting the performances in," said Whelan.
"The more games you play, the better you feel in terms of confidence and also match fitness.
"That helped me because at the start of the season I wasn't playing as many games as I would have liked for Stoke.
"If I am not playing well for Stoke then I don't think I would have a chance with Ireland."
The only blot on Trapattoni's 35-year managerial career is the four-year spell he had as manager of the Italian national team.
South Korea knocked Trapattoni's Italy out of the 2002 World Cup at the last 16 stage, in a match littered with controversial refereeing decisions, and an extra-time winner which led to its scorer, Ahn Jung-Hwan, being initially sacked by his club side Perugia for knocking Italy out of the tournament.
Giovanni Trapattoni celebrated his 70th birthday on St Patrick's Day
Things did not go much better at Euro 2004, when Italy finished third in their group behind Denmark and Sweden, and failed to reach the quarter-finals.
Now Trapattoni is having another crack at management at international level, and he seems to be getting the hang of it.
The Republic of Ireland were unbeaten in the Italian's first six games in charge, and made a good start in their attempts to qualify for a major tournament for the first time in eight years.
Two wins over Georgia, a home victory against Cyprus, and a creditable draw in Montenegro have put the Republic second in Group Eight, level on points with Italy at the top of the table.
Stephen Kelly is another Stoke player who has been given a chance at international level by Trapattoni.
Kelly, 25, made his debut against Chile in May 2006, but had to patient for a regular spell in the national side.
He started the Republic's last qualifier against Georgia at Croke Park, and is expected to keep his place at right-back ahead of Paul McShane for the games against Bulgaria on 28 March and Italy on 1 April.
He said: "They're two massive games for us, and if we get the right results it's going to leave us in such a fantastic position to qualify.
"The way things have gone before in the group, it's been really good. Hopefully we can win at home against Bulgaria and then anything can happen out there in Italy.
Stephen Kelly has won 13 international caps for the Republic of Ireland
"You live for games like that. It's fantastic playing for Stoke and playing in the Premier League, but you go away and play in an international game against Italy, in Italy - you don't get many opportunities like that in your career.
"Hopefully I can play or just be involved over there. That would be one for the scrapbook."
Whelan added: "Bulgaria and Italy are our main rivals in the group but hopefully we can get some points there that will put us in a good position for qualifying.
"If we qualify for the World Cup, it would be one of the highlights of my career.
"Every young lad growing up has dreamed of playing for their country and playing in a World Cup. I'm no different, so fingers crossed."
If Whelan, Kelly and the rest of the Republic of Ireland squad achieve that goal, a lot of the credit will go to Trapattoni.
He would have taken a national football team that had been languishing in the doldrums for far too long, moulded a squad of young, hungry and talented players, and taken the Republic of Ireland back to the biggest stage in world football - the World Cup.
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