Several decades of invasive species research have yielded a broad understanding of the nature of species transfer mechanisms and associated threats globally. This is not true of the Arctic, however, a region where increasing human activity and ongoing climate change is expected to promote species invasion. This thesis examines the potential for both terrestrial and marine non-indigenous species (NIS) to be introduced to and establish in the Arctic under present and future climatic conditions. Throughout, the work uses the high-Arctic archipelago Svalbard as a model for the wider Arctic region. The research focuses on two of the most well-described pathways of species introduction globally, human visitation and shipping, both of which are increasing in intensity in Svalbard. Potential for species introduction and establishment is examined by quantifiying and identifying propagule loads transferred to the Arctic; developing and testing species identification methods; evaluating present and forecasting future habitat suitability for NIS; measuring the spread of established non-indigenous vascular plants; and testing the efficacy of management measures designed to prevent further species introduction. Results demonstrate high plant propagule transport by people travelling to highly-visited Arctic regions is occurring. Furthermore, propagule pressure associated with ship hull fouling poses immediate risks, while if more stringent management related to ships' ballast water discharge is not enacted this vector will pose an increasing risk over the coming century. Improved vector screening methods were achieved through testing a molecular species identification approach for organisms transported with ships, but the approach was found to be inefficient in a biosecurity management context. Climate changes, and particularly temperature increases, over the coming century are expected to increase Svalbard habitat suitability for both terrestrial and marine species. Acknowledgment of the negative impacts NIS may have in Svalbard has led to the implementation of preventative management measures designed to reduce species transfer by visitors and ships; however, these were found to have limited effect. Scope for improved management is outlined. Where species invasion risks are found to exist at the transport stage, the body of invasion ecology knowledge suggests a precautionary approach whereby NIS introduction should be prevented. The imperative to ensure this in polar regions has historically been lacking, owing largely to the strength of climatic barriers, and assumed weak propagule pressure. By quantifying propagule pressure across different pathways and vectors, and estimating changing habitat suitability under forecast climatic conditions, this research provides the basis upon which to develop more informed biosecurity management for Svalbard. Moreover, given the similarity in pathways of species introduction across the Arctic region, the work presented here suggests an Arctic-wide need to address management and policy gaps.