Solving the Dirichlet boundary value problem for an elliptic operator amounts to study the good properties of the associated elliptic measure. In the context of domains having an Ahlfors regular boundary and satisfying theso-called interior corkscrew and Harnack chain conditions (these are respectively scale invariant/quantitative versions of openness and path-connectivity) we will show that for the class of Kenig-Pipher uniformly elliptic operators thesolvability of the Lp-Dirichlet problem with some finite p is equivalent to theuniform rectifiablity of the boundary. Joint work with S. Hofmann, S. Mayboroda, T. Toro, and Z. Zhao.