They have been linked in various studies to weight gain. Why? There are differing reasons. One is that it tricks your brain into wanting regular sugar and the cravings are harder to stave off.

Here is an excerpt from an article on Mark Sisson's website Mark's Daily Apple:
"Another study, which I covered a couple years ago, analyzed the diets of more than 9,500 men and women between the ages of 45 and 64 and found that drinking diet soda was associated with a 34% higher risk of developing metabolic syndrome – the perfect storm of high triglycerides, belly fat, insulin resistance, and obesity that’s so popular nowadays. This was an even stronger association than the one between the “high-meat, high-fat” Western diet and metabolic syndrome.

Authors of both studies speculate that diet soda drinking just extends the life of sugar cravings, rather than eliminating it. In this scenario, diet soda doesn’t regulate the desire for sugar; it increases it, and diet soda drinkers are simply replacing those empty calories with real sugar. This makes sense, and I think it’s part of it, but a couple other studies suggest that something else is going on entirely independent of caloric intake:

The dietary habits and weights of a homogenous group of middle aged women were tracked for a year. Regardless of initial weight status and inexplicable by “food consumption patterns,” users of diet soda were more likely than nonusers to gain weight. They didn’t eat markedly different from non-soda drinkers and yet they got fatter."

Interesting stuff.