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In the movie 'Harry Potter and the Deathly Hallows, Part 1' Hagrid brought the real Harry to the Burrow for William and Fleur's wedding. When the party is attacked, Harry and gang apparate to the coffee bar which is the last time we see Hagrid in that film.

He then shows up next, in 'Harry Potter and the Deathly Hallows, Part 2' he's tied in ropes when Harry comes to confront "you-know-who" in the forest.

When, why and how was Hagrid captured?

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This is made clear in the books (which I don't have access to right now in order to provide a quote), but I guess not the movies. Hagrid was inside the Hogwarts castle when the Battle of Hogwarts began. The giant spiders which live in the Forbidden Forest are driven out by Voldemort and his supporters, and attack the castle. Hagrid gets carried off by them back to the forest, which is when he was captured by Voldemort.

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Hagrid's capture occurs (in the novel, at least) during the Battle of Hogwarts.

‘HAGRID, COME BACK!’
But he was not even halfway to Hagrid when he saw it happen: Hagrid vanished amongst the spiders, and with a great scurrying, a foul swarming movement, they retreated under the onslaught of spells, Hagrid buried in their midst.
‘HAGRID!’
Harry heard someone calling his own name, whether friend or foe he did not care: he was sprinting down the front steps into the dark grounds, and the spiders were swarming away with their prey, and he could see nothing of Hagrid at all.
‘HAGRID!’
He thought he could make out an enormous arm waving from the midst of the spider swarm, but as he made to chase after them, his way was impeded by a monumental foot, which swung down out of the darkness and made the ground on which he stood shudder.
- Harry Potter and the Deathly Hallows

Although we don't see his capture, we can make the assumption that once he was outside the Great Hall he was simply taken by the Death Eaters under Voldemort's command.

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