The other day, I took my son to a matinee of Journey to the Centre of the Earth. (He liked it; I had a chuckle, mainly over the 3D stuntiness of it all.) At the popcorn stand, I was surprised to see a mom with a small boy of about three or four, loading up with snacks tied into the Batman promotion. And yup, into the Dark Knight they went.
I hadn't seen The Dark Knight by that point, but I had already been steered clear by parental warnings ... including one direct note from my friend Mike, who said, "Do NOT take Nicholas to it." I saw the movie, on my own, later on, and have to agree.
The Dark Knight is rated PG, and while it has almost no gore, it's pretty intense. It's a very well-crafted film, and it makes terrific, explosive use of tension, but it's way too much for my eight-year-old ... even though he's seen the preceding film, Batman Begins, on DVD. (Indeed, the film makes some very adult observations about using fear to manipulate people, but that's another kettle of fish.)
I'm not surprised parents would think the film might be appropriate: the stores are filled with child-targeting toys, products and spinoffs. But Christopher Nolan's film is aimed very much at adults, teenagers and mature children.
The choice is up to parents, but I'd stick to the more kid-friendly versions of the Batman story for younger kids.