Donny van de Beek and Matthijs de Ligt scored the goals as Ajax came from behind to beat Juventus in Turin.
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Ajax produced a stunning turnaround to reach their first UEFA Champions League semi-final since 1996/97.
Juventus took the lead when Cristiano Ronaldo met Miralem Pjanić's corner with a thumping header but that advantage only lasted six minutes as Donny van de Beek coolly slotted in the equaliser after Hakim Ziyech's shot deflected into his path.
Wojciech Szczęsny – making his 50th UEFA club competition appearance – made crucial saves from Ziyech and Van de Beek early in the second half but his resistance was finally broken by Matthijs de Ligt's towering header from Lasse Schöne's corner.
View from the stadium: Paolo Menicucci
I have seen Juventus lose in this stadium, albeit not many times, but I have never seen a team dominate here as Ajax did in the second half tonight. The young Dutch side played brilliant football and could have scored many more if they had been more clinical.
This is probably the only thing Ajax coach Erik ten Hag will have to work on. A bit more determination in front of the opposition goal and this almost perfect football machine could go all the way in the competition.
Man of the match: Matthijs de Ligt
A truly remarkable display. Up against one of the game's best strikers in Ronaldo and captain of a young, relatively inexperienced team playing in one of the toughest venues in Europe, he once again shone like a beacon. Dominant in the air and extraordinarily strong in the tackle, De Ligt is confident when distributing the ball. And what about that goal! For someone who isn't yet 20, he is becoming something of a phenomenon.
5: Ajax are now unbeaten in their last five fixtures away to Serie A sides (W3 D2).
7: Ronaldo has scored in his last seven UEFA Champions League quarter-final matches – 13 goals in total.
9: Ronaldo has amassed nine goals in seven appearances against Ajax.
18: Ajax have lost one of their last 18 UEFA Champions League games (including qualifying).
19: De Ligt is the the youngest Dutch player to score in a UEFA Champions League knockout match since 1996.
26: Ajax have failed to find the net in just one of their last 26 UEFA Champions League games.
What does it mean?
Juventus suffer a second successive quarter-final defeat while Ajax have bucked a trend in their history. The Amsterdammers had lost nine of the 12 UEFA competition ties in which they drew the first leg at home, including four European Cup quarter-finals, prior to this comeback.
Juventus require just a point at home to Fiorentina on Saturday to wrap up their eighth straight Serie A title; they then go to Inter in the Derby d'Italia before hosting neighbours Torino.
Ajax, level on points with PSV at the Eredivisie summit, visit Groningen on Saturday and welcome Vitesse to Amsterdam next week.