Adrian, 32, joins Liverpool on a free transfer after his contract with the Hammers came to an end this summer.
The Spaniard is Liverpool's third summer signing and moves to Anfield on the same day that Simon Mignolet ended a six-year spell with the Reds to join Club Brugge.
"Obviously I want to win titles. I come here to fight for all the competitions."
Adrian moved to West Ham from Real Betis in July 2013 and has made 125 Premier League appearances, keeping 35 clean sheets.