The three papers below contain almost no proofs. The first one is a high level introduction, to tell you what this is all about. The Bourbaki seminar goes a little bit more into the mathematics. The ICM talk is a much more complete account of what I knew at the time. It is more impressive, so don't be discouraged. It is also rather obsolete by now.
Spin glasses:a new direction for probability theory ?, Atti Accad. Naz. Lincei Cl. Sc. Fis. Math. Nat. Rendi. Lincei Mat. Appl., 127-146, 2000 (special issue).
Verres de Spin et optimisation combinatoire, Seminar Bourbaki, March 1999 (In French) Asterisque 266, 287-317, 2000.
Huge Random Structures and Mean Field Models for Spin Glasses, Proceedings of the Berlin international Congress of Mathematicians. Documenta Math., Extra Volume I, 1998.
On the other hand, the following introductory material contains proofs in complete details. It has been made obsolete by my book "Spinglasses, a Challenge to Mathematicians", which itself in in the process of beeing made obsolete by the new edition in preparation.
Mean Field models for spin glasses, a first course
to appear in the proceedings of the 2000 Saint Flour Summer School in probability.