The company behind the suit, Between the Lines Productions, is the same one that made a "Twilight" parody film called "TwiHard," and the basis for their suit is that Lionsgate and Summit, "Twilight"'s distributors, tried to block their movie from release with baseless legal threats. But their complaint also contains a multitude of criticisms about the "Twilight" films themselves.
Whatever the motivation, the suit is filled with multiple examples of what Between the Lines dubs rampant racism, sexism, and sexual perversion. One example is the "one-dimensional stereotypes about Native Americans and indigenous culture through the depiction of the character Jacob Black as a 'noble savage,' bloodthirsty warrior' and 'sexual predator."
The complaint also claims that the movies highlight domestic violence, as well as "the lustful and eventual sexual relationship between a 17-year-old girl, Bella Swan, and a male character, Edward Cullen, who is nearly 100 years her senior."
It's unclear whether they also complained that a vampire-human hybrid baby was just plain creepy.