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Holders Chelsea will face 13-time winners Real Madrid in the quarter-finals of the Champions League.
Manchester City will play Atletico Madrid, while Liverpool tackle Benfica and Bayern Munich take on Villarreal.
In the semi-finals, the winner of the Chelsea v Real Madrid tie will face Atletico Madrid or Manchester City, while Benfica or Liverpool will play either Villarreal or Bayern Munich.
The two-leg quarter-finals take place on 5-6 April and 12-13 April.
The first legs of the semi-finals will be on 26-27 April, and the second legs on 3-4 May.
The Stade de France in Paris will host the final on Saturday, 28 May after it was moved from St Petersburg after Russia's invasion of Ukraine.
In the Europa League, West Ham will play Lyon in their first European quarter-final since 1981.
Rangers take on Portuguese side Braga, while Eintracht Frankfurt play Barcelona and RB Leipzig face Atalanta.
The draw means Chelsea face a reunion with their former manager Carlo Ancelotti, who won the Premier League and FA Cup double during a two-year spell with the Stamford Bridge club between 2009 and 2011.
The Italian is one of only three managers to win the Champions League three times, including with Real in 2014, and returned for a second spell at the Spanish giants last summer after leaving Everton.
Chelsea and Real also met at the semi-final stage last season, with Thomas Tuchel's side winning 3-1 on aggregate before going on to beat Manchester City in the final.
The London club have never lost in five previous meetings with Real Madrid.
Atletico Madrid have the chance to complete a Manchester double after being drawn against Pep Guardiola's City, with Diego Simeone's side having beaten Manchester United 2-1 on aggregate in the quarter-finals.
It will be the first competitive meeting between the sides in European competition as City, who thrashed Sporting Lisbon 5-0 on aggregate to reach this stage, seek a first Champions League title.
Liverpool meanwhile have played Portuguese giants Benfica 10 times in Europe, most recently a 5-3 aggregate win in the Europa League quarter-finals in 2010.
Jurgen Klopp's side are seeking an unprecedented quadruple, having already lifted the Carabao Cup and still being in contention for the Premier League and FA Cup.
Chelsea v Real Madrid
Manchester City v Atletico Madrid
Villarreal v Bayern Munich
Benfica v Liverpool
Manchester City or Atletico Madrid v Chelsea or Real Madrid
Benfica or Liverpool v Villarreal or Bayern Munich
West Ham's reward for beating six-time winners Sevilla is a meeting with Ligue 1 side Lyon. It will be just the second time the Hammers have faced French opposition in European competition after beating Metz in the 1999 Intertoto Cup final.
Rangers meanwhile face familiar opposition in Braga having beaten the Portuguese side 4-2 on aggregate in the last 32 of the Europa League in 2020.
Giovanni van Bronckhorst's side reached the last eight thanks to a 4-2 aggregate win against Red Star Belgrade of Serbia.
Should West Ham reach the semi-finals they could meet Barcelona if the Catalan giants overcome Eintracht Frankfurt, while Rangers will face either German club RB Leipzig or Atalanta of Italy if they reach the last four.
The draw means the earliest the two British sides can meet is in the final.
The two-leg quarter-finals will take place on 7 and 14 April, the semi-finals are on 28 April and 5 May with the final in Sevilla's Ramon Sanchez-Pizjuan Stadium on Wednesday, 18 May.
RB Leipzig v Atalanta
Eintracht Frankfurt v Barcelona
West Ham v Lyon
Braga v Rangers
RB Leipzig or Atalanta v Braga or Rangers
West Ham or Lyon v Eintracht Frankfurt or Barcelona
In the Europa Conference League, Leicester's reward for beating French side Rennes is a quarter-final against Dutch side PSV Eindhoven.
The first legs will be played on Thursday 7 April, with the second legs a week later. The two-leg semi-finals are on 28 April and 5 May with the final in Tirana, Albania on Wednesday, 25 May.
Bodo Glimt v Roma
Feyenoord v Slavia Prague
Marseille v PAOK Salonika
Leicester v PSV Eindhoven
Leicester or PSV Eindhoven v Bodo Glimt or Roma
Feyenoord or Slavia Prague v Marseille or PAOK Salonika
