  We review the approach to glasses based on the replica formalism. The replica
approach presented here is a first principle's approach which aims at deriving
the main glass properties from the microscopic Hamiltonian. In contrast to the
old use of replicas in the theory of disordered systems, this replica approach
applies also to systems without quenched disorder (in this sense, replicas have
nothing to do with computing the average of a logarithm of the partition
function). It has the advantage of describing in an unified setting both the
behaviour near the dynamic transition (mode coupling transition) and the
behaviour near the equilibrium `transition' (Kauzmann transition) that is
present in fragile glasses. The replica method may be used to solve simple mean
field models, providing explicit examples of systems that may be studied
analytically in great details and behave similarly to the experiments. Finally,
using the replica formalism and some well adapted approximation schemes, it is
possible to do explicit analytic computations of the properties of realistic
models of glasses. The results of these first-principle computations are in
reasonable agreement with numerical simulations. Draft of a chapter prepared
for the book "Structural Glasses and Supercooled Liquids: Theory, Experiment,
and Applications."
