Brady has left relegated Hull to link up with the newly-promoted Canaries and agreed a three-year contract at Carrow Road.
The 23-year-old is Norwich boss Alex Neil's third recruit of the summer following the arrivals of Graham Dorrans and Youssouf Mulumbu.
Brady has made 43 appearances in the Barclays Premier League over the last two seasons, and has been capped 13 times by the Republic of Ireland.
Neil told the Canaries' official website: "I'm really pleased to sign Robbie
As everyone's aware, we've been chasing him for quite a while.
"He'll be an absolutely perfect fit for us
Hopefully he can go on and show everything that we think he's capable of.
"He's dynamic, he's got good energy to get up and down the pitch, and he's got a fantastic left foot which makes him a real danger at set-pieces
He's got a lot of things in his locker.
"It gives us good competition down the left side
He's got good pedigree and he's still young so he's got his best days ahead of him, so we're just really happy that we've managed to get it over the line."
Brady could make his first appearance in Canary yellow in Saturday's pre-season friendly against Brentford.
Source : PA
Source: PA