There have been many studies looking for a possible link, but a lot of them are small or have problems with their design. However there have been a few larger studies that, while they also have some problems, help to give us a clearer idea of where the evidence is pointing.
The IARC decision in 2011 was mainly based on the results of two sets of studies - research conducted by the Swedish Hardell group and a very large international study called InterPhone - but it's important to remember that IARC didn't feel this was enough hard evidence to come to a decision.
The Hardell studies suggest a link between using a mobile phone and a few specific types of brain tumour, particularly in heavier users. The InterPhone study, which included over 6,000 people across 13 countries, largely found no link between mobiles and brain tumours, apart from in the ten percent of people who used their phones the most - but this could be explained by problems with the study design.