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PSG came into the match unbeaten in their last 27 games in all competitions but were undone by a fired-up Barcelona side looking to return the club to the summit of European football.
Federico Valverde salvaged a draw for Real late on after City had taken a 3-2 lead with sumptuous strikes from Phil Foden and Josko Gvardiol following a Ruben Dias own goal and Rodrygo strike that gave the hosts a half-time lead in Spain's capital.
Islamic State was alleged to have threatened four venues for this week's Champions League matches, including the Parc des Princes in Paris and two games in Madrid hosted by Real Madrid and Atletico Madrid.
Paris St Germain, who are seeking their first Champions League title, will take on Barcelona while Atletico Madrid face Bundesliga side Borussia Dortmund, which means all three Spanish teams have the opportunity to reach the semifinals.
Atletico Madrid needed a penalty shootout to edge past Inter Milan and reach the Champions league quarterfinals
Arsenal survived a penalty shootout to scrape past Porto and reach the quarter-finals of the Champions League for the first time in 14 years.
But Leipzig's Willi Orban levelled the score three minutes later with a header to set up a tense final few minutes that included a strike that hit the crossbar by Dani Olmo.
Paris St Germain's Kylian Mbappe scored twice to lead them into the Champions League quarterfinals.
But the 30-time Portuguese champions will travel with hope.
Real Madrid's Brahim Diaz scored a stunning goal after a mazy run as they won 1-0 at RB Leipzig on Tuesday.