Surface plasmons are able to generate extremely strong and confined optical fields at a deep-subwavelength scale, far beyond the diffraction limit, and now play a central role in nanosciences. A proper combination of plasmonic nanostructures can support Fano resonances arising from the interference between a non-radiative mode and a continuum of radiative electromagnetic waves. Fano resonances are able to confine light more efficiently and are characterized by a steeper dispersion than conventional plasmon resonances, which make them promising for nanoscale biochemical sensing, switching or lasing applications. Unfortunately, these technological developments are hindered by a lack of theoretical and numerical models able to deliver insights into the mechanisms of Fano resonances in plasmonic systems; e.g. to determine the best configuration for specific applications based on this phenomenon. In this thesis, the fundamental properties of Fano resonances in plasmonic nanostructures, and more generally in non-conservative systems, are investigated. An ab initio framework to describe their properties is developed and an analytical formula for their spectral response is derived. An equivalence between the derived resonance formula and the model of two coupled oscillators is also drawn, which confirms the general character of the developed framework. Furthermore, an original surface integral formulation for light scattering by periodic structures is developed and implemented. With this versatile numerical method, a very large variety of geometries can be simulated. The surface discretization using finite elements provides a high flexibility, allowing the investigation of irregular shapes. Thanks to the singularity subtraction technique, insights into the extreme near-field of the scatterers as well as into the corresponding far-field can be obtained with great accuracy. This particular advantage of the surface integral formulation, compared to other numerical methods, enables the detailed study of all the different aspects of Fano resonances in realistic plasmonic systems. The developed theoretical and numerical models are then used to elaborate a methodology to tailor the optical response of plasmonic Fano resonances in the far-field and the near-field. It is also shown that there exist three different coupling regimes in Fano-resonant systems, each regime exhibiting specific properties: in the weak coupling regime, a very high sensitivity to the opening of a radiative channel for the dark mode is observed. An optimal regime of highest electromagnetic field enhancement is obtained only when the in and out-coupling balance intrinsic losses. Finally, for stronger coupling, the specific features of Fano resonances are altered. In the last part, this knowledge of the mechanisms of Fano-resonant plasmonic systems is translated to the optimization of nanoplasmonic systems for a broad range of applications. In the weak coupling regime, radiative losses of the dark mode are almost suppressed and the modulation depth becomes a physical value extremely sensitive to the modes coupling, which can be used for nanoscale plasmon rulers to measure nanometric displacements. In the intermediate regime, the best electromagnetic field enhancement is obtained, which optimizes devices for second harmonic generation, as well as for surface enhanced Raman scattering or antenna-based trapping for biomolecular recognition. The sensitivity of Fano-resonant systems to local perturbations of the refractive index is then discussed. Higher figures of merit than conventional plasmon resonances can be obtained because the contribution of radiative and non-radiative losses to the spectral width can be controlled. This analysis finally leads to the introduction of an intrinsic figure of merit for refractive index sensing using Fano-resonant systems.