There are essential connections between the isoperimetric theory and information theoretic inequalities. In general, the Brunn-Minkowski inequality and the entropy power inequality, as well as the classical isoperimetric inequality and the classical entropy-moment inequality, turn out to be equivalent in some certain sense, respectively. Based on such connections, this thesis is devoted to the investigation of the affine isoperimetry and the establishment of stronger versions of information theoretic inequalities than the classical ones. Gaussian distribution and its various extensions play core roles in the research of probabilistic inequalities and information theoretic inequalities. In this thesis we study a notion of generalized Gaussian distribution by following the work of Lutwak, Yang and Zhang. Such generalized Gaussian distribution not only contains the classical Gaussian as a special case, but also is heavy-tailed and admits to a definition for negative parameters. Information theoretic quantities, which are invariant under the entropy-preserving transformations, are of special interest. This is because such notions are natural to study when there is no "a priori" best or natural choice to define the total error for a given error vector. We will introduce affine "p"th moment for p = 1, affine harmonic moment, and affine Fisher information, of a random vector in [Special characters omitted]. New moment-entropy and Fisher information inequalities that imply the classical ones also are established, in which the generalized Gaussian distributions appear naturally as extremal functions. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]