"I'll probably go to London and hide," he laughed. "Since the day I finished shooting there's been at least one person come up to me every single day and then after the trailer came out, at least four. It's absolutely bizarre to me. This was before there was any systematic promotion of the movie. It's just completely nuts."
Twilight, which is based on the novel by author Stephanie Meyer, has done brisk business in the US Box Office, and looks likely to do well when it is released in the UK next week.
But this latest movie is not Robert's first film adaptation from a bestselling book - he also starred in Harry Potter And The Goblet Of Fire.
"You're not playing the book, you're playing the adaptation otherwise it's just impossible," says Robert, who stars as 'vegetarian' vampire Edward Cullen in Twilight. "The book just becomes material afterwards and you can take bits and leave bits - I shouldn't really say that or people will probably get annoyed with me. I did the same thing with Harry Potter, you just take the bits you like and which relate to your idea of the character and just abandon anything that contradicts that."
Twilight is released in the UK on Friday.