Interest in plasmonic nanostructures has recently multiplied due to their extraordinary optical properties. When resonantly excited, these tiny metallic systems exhibit a strong and localized optical field enhancement. Applications exploiting this effect exist in many fields ranging from thermal cancer therapy to single-molecule spectroscopy. When designing an application, precise knowledge of a structure's optical response is required. The extraordinary behavior of plasmonic systems, however, sets new demands on the used numerical tools. In this thesis, a numerical approach based on the surface integral equation (SIE) formulation is presented. This approach unites the advantages of many popular simulation methods in a single technique and provides the means for in-depth studies on plasmonic structures. It allows for flexible meshing permitting investigations of realistic particles and can accurately describe a structure's optical response in both the far and near-fields. An efficient numerical implementation is presented leading to fast and accurate simulations. Special care is taken to provide enough detail to reproduce the presented method using this work alone. To verify the SIE approach's accuracy, simulations are performed for a system than can be computed analytically. The results show that the SIE produces accurate results and converges more quickly than the volume integral formulation, another popular simulation approach. The validity of the SIE method is assessed and situations in which it might produce errors carefully analyzed. Post-processing of simulated data is often necessary to obtain information that can be compared with experimental results. General formulae for calculating optical cross sections are given and can be significantly simplified when using the SIE approach. Possible error sources of the cross-section calculations are pointed out and a recipe for maximizing the accuracy is given. Furthermore, methods for calculating the effect of a photonic system on a molecule are derived. Though the effects themselves can only be explained using quantum mechanics, they can in fact be quantitatively characterized using the fully classical SIE technique. As SIE-simulations often produce complex field distributions in three dimensions, two unique methods for field representation are described to facilitate their interpretation: a transparent representation for scalar fields such as the intensity and a field line plot for vector fields. Finally, two studies on plasmonic systems using the SIE method are presented. First, the excitation and reemission of molecules in the near-field of plasmonic nanostructures is studied. In particular, the effect of simplifications often performed in numerical simulations is investigated. The need to consider the structure's realistic geometry is demonstrated in cases where the interacting molecule's location is well-defined, such as when combining fluorescence with optical trapping. Properly tuning the resonance of the plasmonic structure is important for obtaining a strong reemission. The optimal configuration, however, is not straightforward and may depend on the properties of the molecule. A procedure for maximizing the measurable signal is presented, optimizing both spectral and geometric properties of the system. A novel application of plasmonic nanostructures is presented last. The strong and very localized field enhancement near a plasmonic dipole antenna leads to a strong field gradient. This gradient can be used to excite atomic dipole-forbidden quadrupole-allowed transitions. Simualtion results show that using a plasmonic nanoantenna, the transition rate of the 62S1/2 – 52D5/2 transition in cesium can be enhanced by many orders of magnitude. Applications of this transition mediation may be found in nanoscale atomic clocks or quantum computing.