The thesis is organized in three parts, each of them in the form of an independent paper. In Part I, titled "A Spectral Theory for Two-Component Porous Media", the spectral representation formalism is applied to successfully explain the observed static (frequency-independent) and dynamic (frequency-dependent) dielectric properties of rock-and-brine systems. In addition, the theory shows scaling properties which were formulated earlier for discrete systems, namely, random resistor networks. A new critical behavior in these porous systems is discovered, whose critical exponent can be related to the transport critical exponents. The exponents describing power law properties of the average dielectric function do not seem to be universal for the continuum porous systems;In Part II, titled "Critical Behavior in a New Random Self-Similar System", we model a continuum porous mixture, in which spherical holes (in any arbitrary euclidean dimension) of one medium are first punched out in the second medium, and, subsequently and iteratively, in the resultant medium of the previous punching. A new iterative Effective Medium Theory (EMT) is developed to represent the average dielectric properties of such a system. The spectral representation formalism is applied to extract geometrical properties, as well as three critical exponents of the dielectric function--conductivity exponent t, take off exponent 1, and touch down exponent u--for systems with two to six euclidean dimensions. Their values are computed through numerical iterations. Also, a new feature in the real part of the dielectric function of the composite is found that suggests a value of zero for the capacitative critical exponent s. Remarkably, some of the properties and exponents are quite different from, and opposite to, those observed in the computer-simulated discrete systems, and, possibly, are signatures of the continuum geometry;In Part III, titled "Exact Critical Exponents of the Dielectric Function for a New Random Self-Similar System", we theoretically predict the existence of critical behavior for the above iterative system, and derive exact and general analytical values for the critical exponents. Then, as examples, both EMT and Maxwell-Garnett Theory are used to obtain theory-dependent expressions for the critical exponents. The computational results of Part II are found to be quite close to the EMT-based theoretical results derived in this part.