Metallic nanostructures interact in complex ways with light, forming the subject of plasmonics and bringing novel physical phenomena and practical applications. The fundamental and practical importance of plasmonics necessitates the development of a multitude of simulation techniques. Surface integral equation (SIE) is a numerical method which is particularly suited for simulating many plasmonic systems. In this thesis, we develop SIE-based numerical methods for plasmonics and use them to study plasmonic systems of interest. Electric and magnetic surface currents are the basic quantities calculated in SIE, and it is appealing to directly compute various physical quantities directly using them. We develop a formalism to compute optical forces and torques, polarisation charges and multipole moments using the surface currents for better accuracy and efficiency. Numerical simulation is all about finding the right balance between accuracy and computational cost. SIE allows to choose this tradeoff in computing the integrals for the simulation matrix. We study the effect of the integration routine on the accuracy of the matrix and propose an optimised recipe for evaluating the integrals. Although this recipe incurs an overhead, we show how it becomes necessary in computing some physical quantities and simulating some systems, and how it allows simulations using a coarser discretisation. One drawback of SIE is that it can only simulate domains for which the response of each domain can be expressed in terms of the Green's function for the domain. Only homogeneous and periodic domains could be dealt with till now, limiting its applicability. We extend SIE to simulate nanostructures embedded in the layers of a stratified medium to partly overcome this restriction, paving the way for further improvements. SIE has the ability to model complex and realistic geometries. We exploit this feature to study the effect of fabrication-induced rounding on nanorods and gap antennae. We show how rounding results in blue shift of resonances, migration of charges from corners to edges to faces, and reduced coupling between nanostructures. The surface current-based formalism to calculate optical forces and torques permits their computation for particles in close proximity. We use this to study the internal forces in compound plasmonic systems, and show the presence of strong internal forces between their components. We also demonstrate surprising features such as force and torque reversal, and circular polarisation-dependent behaviour in achiral systems. We then numerically investigate the possibility of using optical torques to orient and rotate plasmonic nanostructures, relying on surface plasmon resonance, retardation effects and circular polarisation. Polarisation charges contain useful information about the behaviour of plasmonic systems, but there are difficulties in understanding and visualising them. We discuss the complex nature of polarisation charges and suggest various techniques to visualise them in complicated systems in a manner which is easy to understand without loss of information. Finally, we utilise the ability of SIE to compute accurate near fields to study the Raman enhancement in multi-walled carbon nanotubes on coating with metal, and the analogous quenching of Raman signal from silicon substrates.