  These are notes from the lectures of Giorgio Parisi given at the autumn
school "Statistical Physics, Optimization, Inference, and Message-Passing
Algorithm", that took place in Les Houches, France from Monday September 30th,
2013, till Friday October 11th, 2013. The school was organized by Florent
Krzakala from UPMC and ENS Paris, Federico Ricci-Tersenghi from "La Sapienza"
Roma, Lenka Zdeborov\'a from CEA Saclay and CNRS, and Riccardo Zecchina from
Politecnico Torino. The first lecture contains an introduction to the replica
method, along with a concrete application to the computation of the eigenvalue
distribution of random matrices in the GOE. In the second lecture, the solution
of the SK model is derived, along with the phenomenon of replica symmetry
breaking (RSB). In the third part, the physical meaning of the RSB is
explained. The ultrametricity of the space of pure states emerges as a
consequence of the hierarchical RSB scheme. Moreover, it is shown how some low
temperature properties of physical observables can be derived by invoking the
stochastic stability principle. Lecture four contains some rigorous results on
the SK model: the existence of the thermodynamic limit, and the proof of the
exactness of the hierarchical RSB solution.
