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The paper is an extended version of the talk which I gave at the XIXCongresso dell'UMI in Bologna in September 2011. The aim of this paper istwofold: first, to give an overview on the recent development in theclassification of surfaces of general type with $p_g=q=2$; second, to point outsome of the problems that are still open.

On the classification of surfaces of general type with $p_g=q=2$