The electrical transport properties of many smart composites and other technologically important materials are dominated by the connectedness, or percolation properties of a particular component. Predicting the critical behavior of such media near their percolation threshold, where one typically obtains the most interesting and useful material properties, is a formidable challenge, and not well understood from a modeling perspective. Here we report on recent mathematical results on lattice and continuum percolation models of these types of materials. In particular, we have found a direct, analytic correspondence between the critical behavior of transport in two component random media around a percolation threshold, and the critical behavior exhibited by phase transitions in statistical mechanics, such as by the magnetization of a ferromagnet around its Curie point. This correspondence has been used to establish that the critical exponents for DC conduction ear a percolation threshold, for both lattice and continuum systems, satisfy the same scaling relations as their counterparts in statistical mechanics. Underlying our correspondence is an integral representation for the effective conductivity, which has exactly the same mathematical form as a corresponding representation for the magnetization of an Ising ferromagnet. The integral representation applies in general to many classical transport coefficients for two component media, and we have used it to obtain rigorous insulator-conductor composites is investigated here, and even when the inclusions are near to touching, the new bounds give a dramatic improvement over the complex versions of the fixed volume fraction and Hashin-Shtrikman bounds found independently by Milton and Bergman in the late 1970's.