The recent surge of interest in positive characteristic birational geometry provided us with a fairly good understanding of the geometry of threefolds in characteristic p>5 and surfaces in any characteristic, but, so far, little was known beyond these cases. In my talk I will explain what are the main issues with extending latest advances to lower characteristics or higher dimensions, and what are the new phenomena that needs to be understood. The talk will be partially based on a joint project with Christopher Hacon.