The application of gold and silver nanoparticles to areas such as medical research is based on unique optical properties exhibited by some metals. These properties are a direct consequence of localised excitations occurring at visible frequencies known as Surface Plasmon Resonances (SPRs). The exact frequency of an SPR induced in a nanoparticle can be 'tuned' in the optical range by, for example, changing the size of gold and silver nanoparticles, or by varying the relative concentrations of gold and silver within an alloy nanoparticle. Whatever the desired frequency, it is critical that the majority of nanoparticles exhibit the frequency within the resolution limit of the imaging system. The research presented here utilises the high resolution imaging and spectroscopy techniques of (Scanning) Transmission Electron Microscopy ((S)TEM) and Electron Energy Loss Spectroscopy (EELS). It is common practice to analyse the optical properties of alloy nanoparticles using techniques that acquire a single spectrum averaged over multiple particles such as Ultraviolet-Visible (UV-Vis) spectroscopy. However, this technique cannot detect any optical variation between the nanoparticles resulting from compositional change. In this research the author demonstrates through the use of EELS that the SPR can be determined for individual gold/silver alloy nanoparticles, for the purpose of determining the extent of their homogeneity. Importantly, the data presented here suggest dramatic variation in SPR frequency between particles and even within the same particle, indicative of large variations in alloy composition. This puts the assumption that alloying can be scaled down to the nanometre-scale to the test. In order to resolve and extract the SPR in both the pure gold and gold and silver alloy nanoparticles, the author has successfully applied multiple post acquisition techniques such as Richardson-Lucy deconvolution and Principle Component Analysis (PCA) to the EELS Spectrum Imaging (SI) acquisition method. Additionally, the valence band EELS data are supported by complementary electron microscopy techniques; Core loss EELS, Energy Dispersive X-Ray Spectroscopy (EDX) and High Angle Annular Dark Field (HAADF) imaging.