Liverpool were 5-1 aggregate winners over Manchester City in the quarter-finals
Liverpool will play Roma in the Champions League semi-finals, with Bayern Munich facing Real Madrid.
The Reds, who beat Manchester City 5-1 on aggregate in the last round, will host Roma on Tuesday, 24 April with the return leg on Wednesday, 2 May.
Arsenal will face Atletico Madrid in the Europa League semi-finals, while Red Bull Salzburg meet Marseille.
The Gunners will play the first leg at home on Thursday, 26 April, with the second leg a week later on 3 May.
The Europa League final will be in Lyon, France on Wednesday, 16 May, with the winner guaranteed a place in next season’s Champions League.
The Champions League final will be held in Kiev, Ukraine on Saturday, 26 May.
Reaction to Champions League semi-final draw
Arsenal were in trouble in Moscow – Wenger
#ItaliaLovesMichaelOliver – Italian fans back English referee
The Salah derby?
Liverpool, Champions League winners in 2005, have not faced Roma in a competitive match since March 2002.
The Reds’ top scorer Mohamed Salah will face the club he left for £34m in the summer.
Roma are fourth in Serie A – 21 points behind leaders Juventus.
They staged a remarkable comeback to beat Barcelona 3-0 on Wednesday and win their quarter-final on away goals after a 4-4 aggregate draw.
“If anybody thinks this is the easiest draw then I cannot help this person,” said Liverpool boss Jurgen Klopp.
“They obviously didn’t see both games against Barcelona. In the first result, 4-1 was not how the game was – it was Lionel Messi genius against a good side of Roma.
“The second leg was outstanding, it was outstanding what they did. They should have probably won 4-0 or 5-0 – I was really impressed.
“It is just the draw and I know it is very exciting, but it is good because the most important news is we are still in the competition.”
‘One of Europe’s top clashes’
Minutes before the draw was made, Bayern announced Eintract Frankfurt boss Niko Kovac will replace Jupp Heynckes as their coach at the end of the season.
Heynckes is in his fourth spell at Bayern, with his third ending with victory in the 2013 Champions League final.
To repeat that feat, he will have to find a way to overcome defending champions Real in the semi-finals, with the German club at home in the first leg on Wednesday, 25 April and then travelling to Madrid on Tuesday, 1 May.
Real, who have won the competition in three of the past four seasons, reached the last four by coming through a dramatic second leg against Juventus.
Leading 3-0 from the first tie, Zinedine Zidane’s side were pegged back to 3-3 at the Bernabeu, before a last-gasp penalty from Cristiano Ronaldo saw them through.
Zidane has been under pressure this season though, particularly for his team’s league form, with Los Blancos fourth in La Liga and 15 points behind leaders Barcelona.
Real beat Bayern in a dramatic and controversial quarter-final last season –
winning 6-3 on aggregate after scoring three goals in extra time – on their way to lifting the title.
“It’s one of European football’s top clashes. It’s the defending champions against a top side with quality players,” said Heynckes.
Arsenal handed tough draw
Arsenal won their quarter-final against CSKA Moscow 6-3 on aggregate, but were made to work in Thursday’s second leg.
Leading 4-1 from the first tie, the Gunners conceded twice in Moscow and needed late goals from Danny Welbeck and Aaron Ramsey to make progress more comfortable.
Winning the Europa League looks to be the Gunners’ most likely way of qualifying for next season’s Champions League, with Arsene Wenger’s side sixth in the Premier League and 13 points adrift of fourth-placed Tottenham.
Atletico, Europa League winners in 2010 and 2012, are considered favourites to win the competition and are second in the Spanish La Liga, 11 points behind leaders Barcelona and with the best defensive record in the league.
They came through their semi-final against Sporting Lisbon 2-1, losing the second leg 1-0 in Portugal.
What are Liverpool’s chances of winning Champions League?
Data analysts at the Euro Club Index, along with Simon Gleave, head of analysis at Gracenote Sports, have simulated the remaining fixtures and given Liverpool a 60% chance of reaching the final and a 20% chance of winning the competition outright.
They rate Real and Bayern both at 50% to reach the final, but give the Spanish team the edge to win the competition with a 35% chance compared to the Bundesliga club’s 34.5%.
Arsenal are given only a 30% chance of progressing past Atletico Madrid and are at 19% to win the competition.