The BBC reports that a very large Danish study found no link between an increase in cancer incidence and mobile phone use: The study looked 425,000 people, 56,000 of them who have used a cellular phone for 10 years or more, including people who had begun using phones in 1982, the old massively and massively radiating models.
The BBC reports, "They found no evidence to suggest users had a higher risk of tumours in the brain, eye, or salivary gland, or leukaemia."
Retrospective studies that rely on the memory of the person being interviewed are typically considered inferior to clinical studies in which behavior is recorded. But this study relied on calling record from phone companies to analyze use, which raises the bar on accuracy by a large measure.
It is certain that some will argue that even 10 years isn't long enough to track whether cancer develops, but that would be incorrect. In any reasonably large population--and 56,000 is large enough--any slow-forming and slow-growing cancers will start to have manifested themselves in a statistical bulge long before the full bulge would form. That is, if some cancers appear due to a cause in 10 to 30 years, an increased risk will appear in a significant subset of people based on their bodies and particular genetic makeup in advance of 10 years. In a study of 500 or 5,000, you won't see that (unless it's a poison or some other strong-acting direct biological agent); in 56,000, yes, you would, even if it meant 5 or 10 additional people developing a certain form of cancer than expected in that population.
This study doesn't specifically determine whether there is risk to immature brains in adolescents and those younger, and the British government has advised no use of mobile phones by children. That's a harder group to track and understand usage, and it's a highly reasonable caution given some biological studies that need further elaboration.