The geometry of a plasmonic nanostructure determines the charge-current distributions of its localized surface plasmon resonances (LSPR), thereby determining the device’s interactions with external electromagnetic fields. To target specific applications, we manipulate the nanostructure geometry to create different electromagnetic multipole moments, from basic electric and magnetic dipoles to more exotic higher order and toroidal multipoles. The nanoscale nature of the resonance phenomena makes electron beam spectromicroscopy techniques uniquely suited to probe LSPRs over a wide spectral range, with nanoscale spatial resolution. We use electron energy loss spectroscopy (EELS) in a monochromated scanning transmission electron microscope and cathodoluminescence spectroscopy (CL) in a scanning electron microscope to probe the near-field and far-field properties of LSPR. Electric dipoles within triangular prisms and apertures in Sierpiński fractals couple as the generation number is advanced, creating predictable spectral bands from hybridized dipole modes of parent generations with hierarchical patterns of high field intensity, as visualized in EELS. A magnetic dipole moment is engineered using a vertical split ring resonator (VSRR), pushing the limits of nanofabrication techniques. On this nanostructure we demonstrate the calculation of spatially resolved Stokes parameters on the emission of the magnetic dipole mode and a series of coupled rim modes. Coupling of the magnetic dipole mode of four VSRRs in a circular array creates an LSPR mode supporting the lesser-known toroidal dipole moment. We further probe the near-field configuration of this 3D array through tilting under the electron beam in EELS, and the far-field emission through CL of higher order rim modes. We also propose further configurations of five and six VSRRs to strengthen the toroidal dipole moment. All of the data presented herein was analyzed using custom Python code, which provides a unique graphical interface to 3D spectromicroscopy datasets, and a parallelized implementation of the Richardson-Lucy deconvolution algorithm. / Thesis / Doctor of Philosophy (PhD) / Certain types of metallic particles are capable of trapping light on a scale far below that which we can see; their light-trapping properties depend on their material and on their geometry. Using these tiny particles, we can manipulate the behaviour of light with greater freedom than is otherwise possible. In this thesis, we study how we can engineer the geometry of these particles to give predictable responses that can then be targeted towards specific applications. We study a fractal structure with predictable self-similar responses useful for high sensitivity detection of disease or hormone biomarkers; a resonating structure emulating a magnetic response which can be used in the design of unique new materials capable of bending light backwards and cloaking objects from sight; and a combination of these resonators in an array to demonstrate exotic electromagnetic behaviour still on the limit of our understanding.