An animal study that fed 0, 4, 20, 100, 500, 2500, and 5000 mg per kilogram of body weight of aspartame to rats saw lymphoma/leukemia increase in female rats, starting from about twice the risk with 20 mg per kilogram of body weight (a person weighing 75 kilograms or 165 lbs, consuming 1500 mg aspartame, or about 8 cans of diet soda) compared with a control group that was not fed aspartame.
It's strange that he pointed me there when the rest of the article talks about how the most recent HUMAN study shows no association between aspartame consumption and cancer rates, but nevertheless, I'd like to deal with this issue.
Before I get into that, let me give a brief overview of the idea behind non-nutritive or "0 Calorie" sweeteners. Sugar (such as sucrose or fructose -- there are many types of sugars) is perceived as sweet because it interacts with taste buds on our tongue. These taste buds have receptors that temporarily interact with sucrose and other sugars, and when that happens our brain gets a signal that says "oooh, this tastes sweet." Each of these sugars has a varying level of "sweetness" as perceived by the brain depending on how it interacts with these receptors. Evolutionarily, this is an advantage in that sugars are used by our bodies for energy, so being able to detect sugar and have a pleasurable response to it makes sense. But when we consume more Calories than we require and we become obese, this reward response can become counterproductive. One weapon against this is to use compounds that activate our taste buds for "sweet" but without that compound being used for energy. Instead, the compound passes through your digestive tract without being absorbed. That way the food tastes good without making us fat