The lace expansion and its applications : Ecole dEté des Probabilités de Saint Flour XXXIV 2004 The lace expansion is a powerful and flexible method for understanding the critical scaling of several models of interest in probability statistical mechanics and combinatorics above their upper critical dimensions. These models include the self avoiding walk lattice trees and lattice animals percolation oriented percolation and the contact process. This volume provides a unified and extensive overview of the lace expansion and its applications to these models. Results include proofs of existence of critical exponents and construction of scaling limits. Often the scaling limit is described in terms of super Brownian motion.