  The description of gauge theories at strong coupling is one of the
long-standing problems in theoretical physics. The idea of a relation between
strongly coupled gauge theories and string theory was pioneered by 't Hooft,
Wilson and Polyakov. A decade ago, Maldacena made this relation explicit by
conjecturing the exact equivalence of a conformally invariant theory in four
dimensions, the maximally supersymmetric Yang-Mills theory, with string theory
in the AdS5 x S5 background. Other examples of correspondence between a
conformally invariant theory and string theory in an AdS background were
discovered recently. The comparison of the two sides of the correspondence
requires the use of non-perturbative methods. The discovery of integrable
structures in gauge theory and string theory led to the conjecture that the two
theories are integrable for any value of the coupling constant and that they
share the same integrable structure defined non-perturbatively. The last eight
years brought remarkable progress in identifying this solvable model and in
explicitly solving the problem of computing the spectrum of conformal
dimensions of the theory. The progress came from the identification of the
dilatation operator with an integrable spin chain and from the study of the
string sigma model. In this thesis, I present the evolution of the concept of
integrability in the framework of the AdS/CFT correspondence and the the main
results obtained using this approach.
