  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.
