Real rallied to score twice late on but could not find the all important third goal that would have knocked out Dortmund.
Instead the German side progress 4-3 on aggregate, where they will face compatriots Bayern Munich if they avoid defeat to Barcelona.
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Summer 2016 is set to be a very exciting time for football fans, as this is when the long awaited Euro 2016 tournament kicks off.
Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.