This thesis deals with the theoretical study of plasmonic excitations in metallic nanostructures. The main issue that we address is the description of the free-electron gas when the size of metallic structures is of the order of 10 nm, that is comparable with the Fermi wavelength of the gas. These are the typical sizes of the nanoplasmonics structures, that can be fabricated nowadays. The model we propose is the hydrodynamic Drude model, a semiclassical model that describes the free-electron gas in a metal as a Fermi gas subject to the electromagnetic force, as defined by the Navier-Stokes like equation. New in this model is the presence of pressure waves, analogous to sound waves, that give rise to a spatially nonlocal optical response. We provide a theoretical derivation of the hydrodynamic equations, and we point out the main differences between the hydrodynamic model and the classical Drude model, that is commonly used in plasmonics. In particular, we show that the surface charge density has a finite thickness in the hydrodynamic model, and we discuss the correct form of the boundary conditions in the case of no electron spill-out. We present the numerical implementation of the hydrodynamic equations in COMSOL, and we apply this code to the study of a cylindrical nanowire, a cylindrical nanowire dimer, and a bow-tie dimer. The final results reveal the blueshift of the surface plasmon resonances with respect to the ones calculated with the Drude model. In a metallic dimer, much of the electromagnetic energy is confined in the gap between the structures, and this gives rise to the phenomenon of field enhancement. We show that the hydrodynamic model causes the enhancement factors to decrease signifi-cantly. The finite thickness of the surface charge layer allows us to calculate the electric field near sharp tips, where the classical model gives divergent results. We apply this concept to the study of a groove structure for SERS applications, and we evaluate the maximum enhancement factor that is possible to achieve with this structure. Finally, we present a new formulation of the hydrodynamic equation, that has the same form of the ordinary wave equation in the local model. This formulation allows us to study the propagation in plasmonic waveguides in the hydrodynamic model. We calculate the dispersion relations for the cylindrical, V-groove, and L-groove waveguides. We evaluate the ultimate surface mode area for both the V-groove, and the L-groove, that has important implications for the understanding of the Purcell effect in spontaneous emission.