Abstract: When incident radiation resonantly excites localized-surface plasmons of metallic nanoparticles, scattering and the magnitude of the electric near-field are enhanced. This phenomenon, known as localized-surface-plasmon resonance, has witnessed a great deal of attention owing to their increasing number of applications; among them, their instrumentation to sense subwavelength objects, thereby breaking the diffraction limit of standard optical apparatus. After an introduction to the mathematical mechanisms behind plasmonic resonances, I will first discuss a mathematical framework for their application to sense objects beyond the diffraction limit. In a second time, I will discuss a new approach for calculating the surface-plasmon resonances of high-aspect-ratio metallic nanoparticles (of otherwise arbitrary shape). I will show how the use of matched asymptotic expansions allows us to formulate a reduced one-dimensional model of the so-called plasmonic eigenvalue problem and use these solutions to compute the resonant response to the incident electromagnetic radiation.