Metallic nanostructures and their extraordinary optical properties have become of great interest in the last two decades. Their ability to concentrate light into areas much smaller than its wavelength and to increase strongly the electric field intensities led to an immense motivation for scientific research. In addition, it promises new and exciting technological applications, ranging from novel light sources, innovations in photovoltaics, improved optical communication and storage to new sensor devices. The interesting properties of metal nanostructures are linked to optical excitations on their surfaces, so-called surface plasmons (SPs), i.e, resonant collective electron oscillations coupled to an evanescent light field.In this work, fast electrons in a transmission electron microscope are used to probe SPs with unprecedented high spatial resolution. In combination with electron energy loss spectroscopy, a detailed study of the spatial distribution of SPs on the nanometer scale becomes possible. For the particle design electron beam lithography is applied to fabricate nanoparticles of various shapes.SP modes sustained by the flat surface and the edge of silver thin-film (nano)structures are probed and it is shown that plasmonic excitations can be decomposed to surface and edge modes as the two elementary building blocks, each having a distinct dispersion relation. For a silver nanodisk, the complete plasmonic spectrum is mapped, leading to the discovery of a new type of plasmon mode, the so-called breathing mode. Finally, a nanodisk is stepwise morphed into a nanotriangle, and the evolution of film and edge modes is traced. The results suggest that disk modes, characterized by their angular order, can serve as a suitable basis for other nanoparticle geometries. Similar to the linear combination of atomic orbitals in quantum chemistry, a linear combination of plasmonic eigenmodes is introduced to describe plasmon modes in different geometries.