I love a good non-verbal character. Not being able to have your character say something seems like a disadvantage, but really, it opens up a lot of other possibilities.
Take this week's review book, for example. Our protagonist is a Utahraptor. Bakker did not create a world where dinosaurs magically have the ability to talk like humans. Instead, he uses other mechanisms to describe what she is thinking and doing and communicating. Some of these methods are better than others. A clunky example is when he says, "If she could speak like a human, she would say..." A much better example is, "She knows her mate is smart. She trusts that he knows what she knows."
You can also describe non-verbal types of communication, like body language (posture, gestures, gaze, facial expressions, etc.) or narrate their internal monologue.
It's easier to work with non-verbal characters in visual media (like graphic novels, movies, and video games) because you don't have to spend a lot of time describing how they're standing or what their face looks like; you can just show it. I'm quite fond of Chell from the Portal games, because she's non-verbal but still has personality (although it's in a way that you can still project yourself onto her... she's a well-done everyman character). Side note: I learned the other day that Blink from X-Men: Days of Future Past has no verbal lines as a little nod toward Chell (since Blink also uses portals).
Who's your favorite non-verbal character? How do they express themselves best?