These notes arose from three lectures presented at the Summer School on Theoretical
Physics "Symmetry and Structural Properties of Condensed Matter" held in Myczkowce, Poland,
on September 11-18, 2002. We review rigged configurations and the Bethe Ansatz. In the
first part, we focus on the algebraic Bethe Ansatz for the spin 1/2 XXX model and explain
how rigged configurations label the solutions of the Bethe equations. This yields the
bijection between rigged configurations and crystal paths/Young tableaux of Kerov, Kirillov
and Reshetikhin. In the second part, we discuss a generalization of this bijection for the
symmetry algebra $D_n^{(1)}$, based on work in collaboration with Okado and Shimozono.