In this work, we review some of the key issues for designing dielectric and metallic arrays in the diffraction or refraction regimes with main emphasis on left-handed electromagnetism. We first discuss dispersion characteristics of periodic dielectric arrays which are structured on the wavelength scale (photonic crystals for optics and electromagnetic band gaps for microwaves) with special attention paid to propagation and refraction effects. Special attention was also paid to the isotropy properties in the Brillouin zone with the prospects of defining a negative refractive index. Then, we considered metallic structures which permit one to synthesize double-negative media with the goal of pushing their operation frequency into the infrared region. For both classes of microstructures and nanostructures, the technological challenges will be addressed by considering air hole arrays in a high refractive index semiconductor substrate and embedded C-shaped and wire metal arrays patterned on low index substrates.