Rigged configuration bijection and proof of the X = M conjecture for nonexceptional affine types

We establish a bijection between rigged configurations and highest weight elements of a tensor product of Kirillov-Reshetikhin crystals for all nonexceptional types. A key idea for the proof is to embed both objects into bigger sets for simply-laced types An(1) or Dn(1), whose bijections have already been established. As a consequence we settle the X=M conjecture in full generality for nonexceptional types. Furthermore, the bijection extends to a classical crystal isomorphism and sends the combinatorial R-matrix to the identity map on rigged configurations.