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In this thesis, a comprehensive analytical and numerical study of optical non-linear effects in plasmonic metamaterials is presented. The new results reported and described in this work can potentially have a significant impact on our understanding of electromagnetic phenomena in artificial optical materials, and facilitate the design and fabrication of new active optical devices with new or enhanced functionality. Equally important, these results could lead to deeper physical insights into the fundamental properties of these metamaterials. To this end, a new analytical formalism based on the multiple scattering theory has been developed, a theoretical framework that allows one to fully characterise the linear and non-linear electromagnetic properties of arbitrary distributions of metallic nanowires. This formalism is unique in allowing readily retrieval of the spatial distribution of the electromagnetic field both at the fundamental frequency (linear analysis) and the second harmonic (non-linear optical response). The formalism also allows for both frequency- and time-domain investigations. Based on this work, a new software tool with unique features has been implemented and used to achieve a better understanding of the intricate electromagnetic phenomena occurring in nano-structured plasmonic systems. In particular, this tool has been used to design and investigate numerically several new non-linear plasmonic structures and nanodevices with remarkable properties. Amongst them were non-linear plasmonic cavities with high quality factors, plasmonic cavities that support non-linear whispering gallery modes and sub-wavelength non-linear plasmonic sensors with enhanced sensitivity and reduced device volume. Several other plasmonic systems that show tremendous potential for the development of advanced metamaterials-based devices have also been explored. Specifically, it was demonstrated that nano-patterned metasurfaces can be employed to achieve polarisation controlled electromagnetic response in arrays of cruciform apertures and magnetisation induced second harmonic generation in chiralmetallic structures. The numerical investigation of photonic superlattices exhibiting zero effective index of refraction has also been discussed.