Even though the two laugh off the persistent claims of a real-life love affair, that hasn't stopped many from thinking otherwise.
So what's really going on? Their costars have some ideas…
Edi Gathegi, who plays Laurent, tells me the cast and crew have pretty much ignored the gossip.
"I can't even think of those two together at all," he told me during his recent stop in Los Angeles for the launch of Catherine Hardwicke's Twilight: Director's Notebook. "Kristen has a boyfriend! It's onscreen and that's it."
Ashley Greene, aka Alice Cullen, says Pattinson and Stewart are, in a way, victims of their own success.
"Their chemistry is fantastic, but no, they are not in a romantic relationship," Greene said at the Twilight at Midnight DVD launch event at Hollywood's Hot Topic store. "They are good actors [is] basically what it comes down to."
She added, "That's kind of the advantage that we had on Twilight. Everyone has such great chemistry. With as big a cast as it was, and as young a cast as it was, we all just got along so fantastically."
And it sounds like all that feel-good camaraderie is already spilling over into New Moon. Greene said the cast is already getting in some bonding time off the clock. Peter Facinelli, who plays vampire patriarch Dr. Carlisle Cullen, recently had everyone over for dinner.
"He cooked pasta for all of us," Greene said. "We all came to his house and ate and drank and talked. It's a bizarre thing, because we kind of all looked at each other and were like, 'Sooo, what's your life been like?' It was fun."
—Additional reporting by Jennifer Cady & Drusilla Moorhouse