"There is plenty of room at the bottom." This bold and prophetic statement from Nobel laureate Richard Feynman back in 1950s at Cal Tech launched the Nano Age and predicted, quite accurately, the explosion in nanoscience and nanotechnology. Now this is a fast developing area in both science and technology. Many think this would bring the greatest technological revolution in the history of mankind. To understand electronic and optical properties of nanostructures, the following problems have been studied. In particular, intensity of mid-infrared light transmitted through a metallic diffraction grating has been theoretically studied. It has been shown that for s-polarized light the enhancement of the transmitted light is much stronger than for p-polarized light. By tuning the parameters of the diffraction grating enhancement can be increased by a few orders of magnitude. The spatial distribution of the transmitted light is highly nonuniform with very sharp peaks, which have the spatial widths about 10 nm. Furthermore, under the ultra fast response in nanostructures, the following two related goals have been proved: (a) the two-photon coherent control allows one to dynamically control electron emission from randomly rough surfaces, which is localized within a few nanometers. (b) the photoelectron emission from metal nanostructures in the strong-field (quasistationary) regime allows coherent control with extremely high contrast, suitable for nanoelectronics applications. To investigate the electron transport properties of two dimensional carbon called graphene, a localization of an electron in a graphene quantum dot with a sharp boundary has been considered. It has been found that if the parameters of the confinement potential satisfy a special condition then the electron can be strongly localized in such quantum dot. Also the energy spectra of an electron in a graphene quantum ring has been analyzed. Furthermore, it has been shown that in a double dot system some energy states becomes strongly localized with an infinite trapping time. Such states are achieved only at one value of the inter-dot separation. Also a periodic array of quantum dots in graphene have been considered. In this case the states with infinitely large trapping time are realized at all values of inter-dot separation smaller than some critical value.