The Arctic region has warmed more than twice as fast as the global average -- a phenomenon known as Arctic amplification. The rapid Arctic warming has contributed to dramatic melting of Arctic sea ice and spring snow cover, at a pace greater than that simulated by climate models. These profound changes to the Arctic system have coincided with a period of ostensibly more frequent extreme weather events across the Northern Hemisphere mid-latitudes, including severe winters. The possibility of a link between Arctic change and mid-latitude weather has spurred research activities that reveal three potential dynamical pathways linking Arctic amplification to mid-latitude weather: changes in storm tracks, the jet stream, and planetary waves and their associated energy propagation. Through changes in these key atmospheric features, it is possible, in principle, for sea ice and snow cover to jointly influence mid-latitude weather. However, because of incomplete knowledge of how high-latitude climate change influences these phenomena, combined with sparse and short data records, and imperfect models, large uncertainties regarding the magnitude of such an influence remain. We conclude that improved process understanding, sustained and additional Arctic observations, and better coordinated modelling studies will be needed to advance our understanding of the influences on mid-latitude weather and extreme events.