In this project, we present the investigation of the optical properties of dielectric/metal core/shell spherical nanostructures by employing the Mie theory. In this work, the nanomaterials, historical background, definition, synthesis, characterization, classification, its special type are covered especially metal core/shell which is the focus of this project is given a special attention. In addition to this one of the physicochemical properties of metal nanostructures and their application, the effects of size, shape and structure on the nanomaterials properties are presented and the question why we use nanoparticles is answered. The other most important point, complex dielectric function of nanomaterials or nanostructures and its relation with the bulk or parent material and plasmon and local surface plasmon resonance of nanostructures are discussed. Topic which leads us to the final goal of this project Mie scattering theory is explained. In addition, the relevant formulas used for the study such as scattering coefficients and absorption, scattering and extinction cross-sections are presented from Mie theory and the extended Mie theory. Finally, using the Mie theory calculator for the silica (SiO2) core, silver (Ag) shell nanostructure and silica (SiO2) core with silver (Ag), gold (Ag) and copper (Cu) shell nanostructures, the cross-sections are calculated and we found that keeping the SiO2 core diameter (50 nm) constant and changing the shell thickness (3-12 nm) and shell type between Ag, Au and Cu cause the spectra to shift towards the longer or shorter wavelength side and also change the absorption, scattering and extinction cross-sections peak, the amount of absorption and scattering cross-sections, whether absorption or scattering cross-sections contributes more for extinction spectra, the wavelength at which these cross-sections occur and the possible application of the nanoshells.