This thesis is about the intriguing phenomenon of extraordinary transmission through metallic sheets periodically nanostructured with subwavelength apertures. An effect related with the excitement of a type of surface waves such as surface plasmons. Furthermore, in structures formed by stacking two or more metallic sheets, new features appear, such as artificial magnetism which gives rise to magnetic resonances and therefore the possibility to get a negative refractive index. By means of theoretical and numerical study, this type of effective magnetic response has been proved to be due to the excitation of plasmonic resonances in the internal structure. Retrieving, under normal incidence, an effective negative refractive index in the direction of propagation in the case of that these resonances occur within frequency regions where negative permittivity is obtained, connecting the world of plasmonics with that of metamaterials. One of the main goals in the design of metamaterials is to achieve a negative refractive index over a broad bandwidth. However, this objective is usually difficult to accomplish owing to most of the designs rely on resonant phenomena. That is why in this thesis, a design based on stacking fishnet structures with different dielectric thicknesses has been proposed to increase the bandwidth at which a negative index is achieved. Basically, obtaining this effect is based on the excitation of plasmonic resonances at different frequencies as the unit cell is made up of different dielectric thicknesses. The hybridization effect that occurs among these resonances permits negative index bandwidth to increase. Although the extraordinary transmission is primarily related to the excitation of surface plasmons, the results shown in the thesis demonstrate that in the case of metallic sheets surrounded by dielectric extraordinary transmission is also achieved due to the coupling of the impinging light to the modes supported by the dielectric media as long as the metal is periodically patterned. These results demonstrate the important role of the periodicity in the onset of the extraordinary transmission phenomenon not only through the excitation of surface plasmon but also dielectric modes. Finally, experimental transmission results are shown through several nanostructured monolayer metallic structures, fabricated varying design parameters such as dielectric media, aperture size and lattice, in order to study its influence on the spectral response. Specifically, on the resonant frequency, quality factor and transmission level. All these investigations were made with the aim of using these structures as ultracompact optical components in future applications based on nano-optical systems. Applications such as, among others, detection, light focusing, or as the presented in the thesis of filtering and optical switching.