The 22-year-old Norwegian has struggled to make an impact in Real manager Zinedine Zidane’s squad this season.
Real Sociedad, where Odegaard spent last season on loan, and Eredivisie leaders Ajax were also interested.
Arsenal boss Mikel Arteta said:
Martins of course a player we all know very well. Although still young, he has been playing at the top level for a while. He would a quality offensive option.
Odegaard speaking about Arteta:
He seems like a top manager and I liked his ideas, the way he sees football and also the way he is. He gave me a great feeling and that was important for me to come here. He was crucial.I think it’s a club that really suits me well.
His arrival at Emirates Stadium comes after the departure of playmaker Mesut Ozil to Fenerbahce.