North American (NA) surface air temperatures in winter have been suggested to be associated with stratospheric variability, such as sudden stratospheric warmings (SSWs) or extreme stratospheric waves. However, the robustness and underlying mechanisms are not well understood. In particular, further studies are needed to better understand the dynamical processes underlying extreme stratospheric wave events. Yet, analysis is hindered by the limited sample sizes and the difficulty of identifying these wave events. In this dissertation, we show that extreme stratospheric wave activity is accompanied by subseasonal fluctuations between warm and cold spells over North America with reanalyses and a hierarchy of climate models. Our study identifies a robust precursor of strong stratospheric wave activity for NA cold extremes on subseasonal timescales. Our findings shed light on the dynamical processes underlying extreme stratospheric wave variability, highlighting the role of vertical wave structure in stratosphere-troposphere coupling.First, we demonstrate that the vertical coupling of extreme stratospheric wave activity is distinct from the well-known anomalous polar vortex events. We measure the stratospheric wave activity using empirical orthogonal function (EOF) analysis of 10 hPa geopotential height. In contrast to the increased persistence of weather regimes following SSWs, we show that extreme stratospheric wave events feature weather transitions between warm and cold spells over North America in reanalyses and climate models with various configurations. Particularly, strong stratospheric wave events are followed by an increased risk of cold extremes over North America 5–25 days later. The NA coldness is more robust following strong stratospheric wave activity than a weak polar vortex. We further examine the causality between stratospheric wave activity and NA cold extremes with idealized nudging experiments in a climate model with a well-resolved stratosphere. The stratosphere in the nudging run is fully relaxed to its counterpart in a free-running control simulation. By comparing the strong wave events between the two runs, we attribute the observed NA cold anomalies to the strong stratospheric wave activity. Moreover, vertical wave coupling is found to be key to the temperature transition during strong wave events. Further examinations of Coupled Model Intercomparison Project Phase 6 (CMIP6) reveal large uncertainty in the wave event evolution across individual models. It is found that models with a degraded representation of stratospheric wave structure also show biases in the troposphere during strong wave events. In order to investigate the role of the Quasi-biennial Oscillation (QBO) in the linkage between extreme stratospheric wave activity and NA temperature, we compare strong wave events during the westerly phase (wQBO) with those during the easterly phase (eQBO). We show that, in contrast to strong stratospheric wave events under wQBO, strong wave events under eQBO do not change the cold risk over North America nor alter the vertical wave structure in the observation. We further examine this QBO dependence in QBO-resolving CMIP6 models, finding that the strong wave events in models are largely insensitive to QBO phases, a possible bias in numerical models.