Eigenmodes are central to the study of resonant phenomena in all areas of physics. However, their use in nano-optics seems to have been hindered and delayed for various reasons. First, due to their small size, the response of nanostructures to a far-field optical excitation is mainly dipolar. Thus, preliminary studies of nanosystems through optical methods meant that only very few eigenmodes of the system were probed, and a complete eigenmode theory was not required. Second, rigorously defining eigenmodes of an open and lossy cavity is far from trivial. Finally, only few geometries allow for an analytical solution of Maxwellâ s equations that can be expressed in terms of modes, rendering the use of numerical methods mandatory to study non-trivial shapes. On the other hand, modern spectroscopy techniques based on fast electron excitation, instead of optical excitation, allow going beyond the above-mentioned dipolar regime and enable the observation of high order modes. In addition, the generation of second harmonic light (SHG) by nanoparticles permits revealing higher order modes that weakly couple to planewave far-field probing. Thus, to be able to analyze the data collected with such experimental methods and comprehend them in order to make appropriate nanostructure designs, one needs to develop suitable numerical tools for the computation of eigenmodes. This is the focus of this thesis, where eigenmodes are used throughout to analyze and understand experimental and numerical results. First, different approaches used to define and compute eigenmodes are presented in details together with the surface integral equation method used in this manuscript. The second chapter presents the use of eigenmodes to study the SHG in plasmonic nanostructures. A single mode is used as an SHG source to disentangle the modal contributions from different SHG channels. For three different nanostructures, the dipolar mode gives a pure quadrupolar second harmonic (SH) response. Then, the interplay of dipolar and quadrupolar SH radiations in nanorods of different sizes is revealed through a multipolar analysis, explaining the experimental observation of the flip between forward and backward maximum SH emissions. Finally, the dynamics of the SHG from a silver nanorod generated by short pulses is investigated. By tuning the spectral position and width of the pulses, the dynamics of a single mode is observed, both in the linear and SH responses, and fits extremely well with a harmonic oscillator model. The last chapter presents the utilization of the eigenmodes to interpret electron energy loss spectroscopy (EELS) measurements. An alternative approach to compute EELS signal is presented, revealing the different paths through which the energy of the electron is dissipated. Instead of computing the work done by the electron against the scattered electric field, the Ohmic and the radiation losses are evaluated. Then, heterodimers with several shapes and compositions are studied. A rich variety of modes is found, due to the additional degree of freedom associated with the different metals. Dolmen shaped nanostructures are also investigated in great details. A rigorous analysis of the eigenmode evolution when the central horizontal nanorod is moved is performed. Finally, we study the EELS for three iterations of a Koch snowflake nanoantenna. The evolution of the modes with the iteration of the fractal is analysed and the modes are linked to the experimental EELS map