In this talk, we study the Ahlfors regularity of planar self-affine sets under natural 
conditions: strong separation condition, strong irreducibility and proximality. Not 
surprisingly, if the dimension is strictly larger than 1, the set is never Ahlfors regular. 
In case if the dimension is less than or equal to 1 under the extra condition of dominated 
splitting, we show that the Ahlfors regularity is equivalent to the positive proper 
dimensional Hausdorff measure and to positive proper dimensional Hausdorff measure of the 
projections in every Furstenberg direction. Moreover, we introduce a condition called 
"projective separation", which is equivalent to having positive measure, and which is 
applicable to show that both positive and zero Hausdorff measure happens on reasonably 
large sets of translation parameters. This is a joint work with Antti Käenmäki and Han Yu.