The Europa League semi-finalists are Manchester United, Celta Vigo, Ajax and Lyon, though three out of four ties went to extra time.
Celta Vigo were the only side to secure their spot in the final four in regular time thanks to a 1-1 draw away to Genk, going through 4-3 on aggregate.
Pione Sisto pounced on a defensive error and smashed in from outside the box, making Leandro Trossard’s equaliser irrelevant.
The Europa League semi-finalists are Manchester United, Celta Vigo, Ajax and Lyon, though three out of four ties went to extra time.
Celta Vigo were the only side to secure their spot in the final four in regular time thanks to a 1-1 draw away to Genk, going through 4-3 on aggregate.
Pione Sisto pounced on a defensive error and smashed in from outside the box, making Leandro Trossard’s equaliser irrelevant.
Manchester United had been considered the favourites to lift this trophy, but made surprisingly hard work of it against Anderlecht by repeating the 1-1 draw from the first leg.
Henrikh Mkhitaryan broke the deadlock early doors, but it was cancelled out when a deflected Youri Tielemans strike came off the bar and into the path of Sofiane Hanni.
The hosts wasted numerous scoring opportunities, several of them with Zlatan Ibrahimovic before he went off with a potentially serious knee injury.
Marouane Fellaini was penalised for climbing on a defender, so the Paul Pogba goal was disallowed, but the Belgian learned his lesson and knocked down a long Daley Blind ball for Marcus Rashford to complete a cool finish.
Anthony Martial had the ball in the net for a third, but only after being caught offside.
Schalke had to recover from a 2-0 first leg defeat to the Ajax youngsters and did it to force extra time with Leon Goretzka and Guido Burgstaller.
The Dutch side went down to 10 men, as Joel Veltman finished off a dismal performance with his second bookable offence for a body-check.
Despite the extra man and a header from Caligiuri, Schalke saw Ajax score in extra time, Nick Viergever bravely sticking out a leg on a set play to fire in from close range. This put the Dutch side ahead on away goals, 3-3 on aggregate.
The Germans folded and were caught out on the counter by Amin Younes to make it an emphatic 4-3 aggregate victory. That was their first passage into a European semi-final since 1996-97.
Besiktas and Lyon played out an extraordinary game in Istanbul that could comfortably have had eight or nine goals each.
Anderson Talisca bagged a brace around Alexandre Lacazette’s sensational lob, but Lacazette hit the woodwork twice, Ricardo Quaresma thumped the post in the final minutes and Ryan Babel missed a sitter.
Besiktas goalkeeper Fabricio was at fault for a goal in the first leg, which ended 2-1 as well, but he made up for it in extra time with an astounding reaction save on Corentin Tolisso.
It could only be decided on a penalty shoot-out and even then all 10 were converted, forcing it to sudden death. Dusan Tosic saw his spot-kick parried by Anthony Lopes, but Jallet ballooned the decisive effort over the bar.
Lopes saved again on Mitrovic and this time captain Maxime Gonalons kept his cool to convert the 16th penalty of the shoot-out and send Lyon through to the semi-finals.
Besiktas 2-1 Lyon (aet, 3-3 agg)
Talisca 27, 58 (B), Lacazette 34 (L)
Lyon win 6-7 on penalties
Genk 1-1 Celta Vigo (3-4 agg)
Sisto 63 (C), Trossard 67 (G)
Man Utd 2-1 Anderlecht (aet, 3-2 agg)
Mkhitaryan 10 (M), Hanni 32 (A), Rashford 107 (M)
Schalke 3-2 Ajax (aet, 3-4 agg)
Goretzka 53 (S), Burgstaller 56 (S), Caligiuri 101 (S), Viergever 111 (A), Younes 120 (A)
Sent off: Veltman 80 (A)