  A comprehensive review will be given about the rich mathematical structure of
mean field spin glass theory, mostly developed, until now, in the frame of the
methods of theoretical physics, based on deep physical intuition and hints
coming from numerical simulation. Central to our treatment is a very simple and
yet powerful interpolation method, allowing to compare different probabilistic
schemes, by using convexity and positivity arguments. In this way we can prove
the existence of the thermodynamic limit for the free energy density of the
system, a long standing open problem. Moreover, in the frame of a generalized
variational principle, we can show the emergency of the Derrida-Ruelle random
probability cascades, leading to the form of free energy given by the
celebrated Parisi \textit {Ansatz}. All these results seem to be in full
agreement with the mechanism of spontaneous replica symmetry breaking as
developed by Giorgio Parisi.
