Holders Real Madrid will face rivals Atletico Madrid in the semi-finals of the Champions League - a repeat of last year’s final.
French side Monaco take on Italian club Juventus in the other last-four tie.
The first legs will be played on 2 and 3 May, with the return legs taking place the following week.
Real, aiming to win Europe’s premier club competition for a 12th time, beat German champions Bayern Munich 6-3 on aggregate to reach the last four.
Atletico, meanwhile, ended Premier League champions Leicester’s fairytale run in Europe, edging the Foxes 2-1 over two legs.
Juventus claimed an impressive 3-0 aggregate win over Barcelona while Monaco defeated Borussia Dortmund 6-3.
In the Europa League, Manchester United will face Spanish side Celta Vigo in the semi-finals of the Europa League.
The Premier League side have won the European Cup three times but have never triumphed in the continent’s secondary club competition.
Dutch side Ajax, winners in 1992, come up against French club Lyon in the other last-four tie.
The first legs will be played on 4 May with the second leg on 11 May, and the final in Stockholm, Sweden on 24 May.
United manager Jose Mourinho won the competition with Porto in 2003 and comes up against La Liga club Celta, who have never won a major European competition.
Ajax, who beat Schalke 4-3 on aggregate to reach the last four, are also experienced European campaigners. They have won the Champions League/European Cup four times previously and the Europa League/Uefa Cup once, beating Torino in the 1992 final.