Ajax reached their first European semi-final in 20 years after beating Schalke 4-3 on aggregate in the Europa League after extra-time.
Leon Goretzka and Guido Burgstaller scored in the space of three second-half minutes to cancel out Ajax's 2-0 advantage from the first leg, before Daniel Caligiuri hit Schalke's third after 101 minutes thanks to Sead Kolasinac's fine cross.
But despite having Joel Veltman sent off in the 80th minute, Ajax replied with a scrambled goal from Nick Viergever and a clinical effort from Amin Younes in the last 10 minutes of extra-time to bring it back to 3-2 and advance on aggregate.
Celta Vigo reached a European semi-final for the first time after a 1-1 draw at Genk gave them a 4-3 aggregate win.
Pione Sisto put the Spaniards ahead on the night with a fierce drive in the 63rd minute after a barnstorming run from the halfway line.
Leandro Trossard levelled just four minutes later but it was not enough to dent the Spaniards' charge to the last four.
In Istanbul, Lyon beat Besiktas 7-6 on penalties after the Turkish side had won 2-1 on the night to level the tie at 3-3.
Matching each other penalty for penalty until number six, Dusko Tosic saw his effort saved by Lyon's Anthony Lopes, but Christophe Jallet skied over the crossbar for Lyon.
Matej Mitrovic's effort was then saved for Besiktas and Maxime Gonalons made no mistake to put Lyon through.
Earlier, Talisca had scored either side of a strike from Alexandre Lacazette to level the tie on aggregate.
Manchester United also reached the last four after beating Anderlecht in extra-time.