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The focus of this paper is to introduce novel, artificial periodic magneto-dielectric meta-material structures that exhibit new electromagnetic functionality and previously unobtainable figures of merit. The main objectives are to accurately characterize the meta-materials and present the behaviors of electromagnetic waves within such composite structures. The powerful computational techniques based on the anisotropic effective medium theory (AEMT), and finite difference time domain (FDTD) broadband analysis with periodic boundary condition/perfectly matched layer (PBC/PML) are applied to characterize the periodic materials. The obtained results can be integrated into the design of novel materials with potential applications in wireless systems. Some applications of meta-materials in the areas of electromagnetic band-gap (EBG) structures and antenna miniaturization are highlighted in this work.