Tickets for the Alhambra are available on the day but they are limited and you need to get there very early in the morning! 8am in April there was already a queue. It's good to get there early anyway as there are less people then.
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The easiest way to get tickets would be to prebook at www.alhambratickets.com. However, when I tried this two weeks ahead of going, it was already all sold out.
So I had to queue on the day like everyone else. I was advised to go an hour before the ticket office opens (it opens at 8am) and it turned out to be a good tip. At 7.15am there were about 25 people in the queue before me. By 8am the queue had grown to maybe 500 people!
There's one more alternative - and a good one actually. If you go to the ticket office as late as 5pm, you won't find anyone queuing at all! During the summer months, the Alhambra is open until 8pm so there's plenty of time left for a good long visit.